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Extending Pascal and Narayana by Product of Fractions Method    

The well-known Pascal's triangle and Narayana triangle can be generated by a variety of methods. This page shows a well-known construction that can be used for Pascal's triangle, and easily adapted to Narayana by first noting that the binomial coefficients C(1,k) for the natural numbers 0, 1, 2, 3, ...; and then treating C(1,k) as a diagonal of the triangle being constructed, i.e. T(1,k). After constructing Pascal and Narayana triangles this way, it is clear that the method extends to higher "levels" of triangles.

The Axioms

These are the rules for constructing a triangle of "level" L, where L=1 gives Pascal's triangle and L=2 gives the Narayana triangle:

Using just these rules there a single wyay to build a triangle for any natural number L.

Illustration

To illustrate the construction, here we create the Narayana triangle which is level L=2.

Set all T(k,0) and T(k,k) to 1: 1 1 1 1 1 1 1 1 1 1 1

Set the elements T(k,1) for k>=1 equal to binomial coefficients C(1+k,2): 1 1 1 1 3 1 1 6 1 1 10 1 1 15 1

Complete the T(3,k) row by filling in the value of T(3,2) using T(r,k) = T(r,k-1)×T(r+1-k,1)/T(k,1).

T(3,2) = T(3,1)×T(2,1)/T(2,1) = 6×3/3 = 6. 1 1 1 1 3 1 1 6 6 1 1 10 1 1 15 1

Complete the T(4,k) row by filling in the values of T(4,2) and T(4,3) using T(r,k) = T(r,k-1)×T(r+1-k,1)/T(k,1).

T(4,2) = T(4,1)×T(3,1)/T(2,1) = 10×6/3 = 20.

T(4,3) = T(4,2)×T(2,1)/T(3,1) = 20×3/6 = 10. 1 1 1 1 3 1 1 6 6 1 1 10 20 10 1 1 15 1

The construction continues in the same manner.

The 1st triangle in the series is Pascal's triangle.

Pascal's triangle (L=1)

1 1
1 1 2
1 2 1 4
1 3 3 1 8
1 4 6 4 1 16
1 5 10 10 5 1 32
1 6 15 20 15 6 1 64
1 7 21 35 35 21 7 1 128
1 8 28 56 70 56 28 8 1 256
1 9 36 84 126 126 84 36 9 1 512
1 10 45 120 210 252 210 120 45 10 1 1024
1 11 55 165 330 462 462 330 165 55 11 1 2048
1 12 66 220 495 792 924 792 495 220 66 12 1 4096
1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 8192
1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1 16384

Diagonals:

T(n+0,0) = 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A0012

T(n+1,1) = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ... A0027

T(n+2,2) = 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, ... A0217

T(n+3,3) = 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, ... A0292

T(n+4,4) = 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, ... A0332

T(n+5,5) = 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, ... A0389

T(n+6,6) = 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, ... A0579

T(n+7,7) = 1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 11440, 19448, 31824, 50388, 77520, 116280, 170544, 245157, 346104, 480700, 657800, ... A0580

T(n+8,8) = 1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, 43758, 75582, 125970, 203490, 319770, 490314, 735471, 1081575, 1562275, 2220075, ... A0581

T(n+9,9) = 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, 92378, 167960, 293930, 497420, 817190, 1307504, 2042975, 3124550, 4686825, 6906900, ... A0582

Row sums: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, ... A0079 (the powers of 2)

Triangle 2, based on seed terms:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, ...

1 1
1 1 2
1 3 1 5
1 6 6 1 14
1 10 20 10 1 42
1 15 50 50 15 1 132
1 21 105 175 105 21 1 429
1 28 196 490 490 196 28 1 1430
1 36 336 1176 1764 1176 336 36 1 4862
1 45 540 2520 5292 5292 2520 540 45 1 16796
1 55 825 4950 13860 19404 13860 4950 825 55 1 58786
1 66 1210 9075 32670 60984 60984 32670 9075 1210 66 1 208012
1 78 1716 15730 70785 169884 226512 169884 70785 15730 1716 78 1 742900
1 91 2366 26026 143143 429429 736164 736164 429429 143143 26026 2366 91 1 2674440
1 105 3185 41405 273273 1002001 2147145 2760615 2147145 1002001 273273 41405 3185 105 1 9694845

Diagonals:

T(n+0,0) = 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A0012

T(n+1,1) = 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, ... A0217

T(n+2,2) = 1, 6, 20, 50, 105, 196, 336, 540, 825, 1210, 1716, 2366, 3185, 4200, 5440, 6936, 8721, 10830, 13300, 16170, ... A2415

T(n+3,3) = 1, 10, 50, 175, 490, 1176, 2520, 4950, 9075, 15730, 26026, 41405, 63700, 95200, 138720, 197676, 276165, 379050, 512050, 681835, ... A6542

T(n+4,4) = 1, 15, 105, 490, 1764, 5292, 13860, 32670, 70785, 143143, 273273, 496860, 866320, 1456560, 2372112, 3755844, 5799465, 8756055, 12954865, 18818646, ... A6857

T(n+5,5) = 1, 21, 196, 1176, 5292, 19404, 60984, 169884, 429429, 1002001, 2186184, 4504864, 8836464, 16604784, 30046752, 52581816, 89311761, 147685461, 238369516, 376372920, ... A108679

T(n+6,6) = 1, 28, 336, 2520, 13860, 60984, 226512, 736164, 2147145, 5725720, 14158144, 32821152, 71954064, 150233760, 300467520, 578399976, 1075994073, 1941008916, 3405278800, 5824819000, ... A134288

T(n+7,7) = 1, 36, 540, 4950, 32670, 169884, 736164, 2760615, 9202050, 27810640, 77364144, 200443464, 488259720, 1126753200, 2478857040, 5226256926, 10606227291, 20796524100, 39525557500, 73018266750, ... A134289

T(n+8,8) = 1, 45, 825, 9075, 70785, 429429, 2147145, 9202050, 34763300, 118195220, 367479684, 1057896060, 2848181700, 7229999700, 17420856420, 40067969766, 88385227425, 187746398125, 385374185625, 766691800875, ... A134290

T(n+9,9) = 1, 55, 1210, 15730, 143143, 1002001, 5725720, 27810640, 118195220, 449141836, 1551580888, 4936848280, 14620666060, 40648664980, 106847919376, 267119798440, 638337753625, 1464421905375, 3237143159250, 6917263803450, ... A134291

Row sums: 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, ... A0108

Triangle 3, based on seed terms:

1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, ...

1 1
1 1 2
1 4 1 6
1 10 10 1 22
1 20 50 20 1 92
1 35 175 175 35 1 422
1 56 490 980 490 56 1 2074
1 84 1176 4116 4116 1176 84 1 10754
1 120 2520 14112 24696 14112 2520 120 1 58202
1 165 4950 41580 116424 116424 41580 4950 165 1 326240
1 220 9075 108900 457380 731808 457380 108900 9075 220 1 1882960
1 286 15730 259545 1557270 3737448 3737448 1557270 259545 15730 286 1 11140560
1 364 26026 572572 4723719 16195608 24293412 16195608 4723719 572572 26026 364 1 67329992
1 455 41405 1184183 13026013 61408347 131589315 131589315 61408347 13026013 1184183 41405 455 1 414499438
1 560 63700 2318680 33157124 208416208 614083470 877262100 614083470 208416208 33157124 2318680 63700 560 1 2593341586

Diagonals:

T(n+0,0) = 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A0012

T(n+1,1) = 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, ... A0292

T(n+2,2) = 1, 10, 50, 175, 490, 1176, 2520, 4950, 9075, 15730, 26026, 41405, 63700, 95200, 138720, 197676, 276165, 379050, 512050, 681835, ... A6542

T(n+3,3) = 1, 20, 175, 980, 4116, 14112, 41580, 108900, 259545, 572572, 1184183, 2318680, 4331600, 7768320, 13441968, 22535064, 36729945, 58373700, 90684055, 138003404, ... A47819

T(n+4,4) = 1, 35, 490, 4116, 24696, 116424, 457380, 1557270, 4723719, 13026013, 33157124, 78835120, 176729280, 376375104, 766192176, 1498581756, 2828205765, 5168991135, 9177226366, 15870391460, ... A107915

T(n+5,5) = 1, 56, 1176, 14112, 116424, 731808, 3737448, 16195608, 61408347, 208416208, 644195552, 1837984512, 4892876352, 12259074816, 29115302688, 65937597264, 143107211709, 298915373064, 603074875480, 1178943365600, ... A140901

T(n+6,6) = 1, 84, 2520, 41580, 457380, 3737448, 24293412, 131589315, 614083470, 2530768240, 9386849472, 31803696288, 99604982880, 291153026880, 800670823920, 2085276513474, 5172303508911, 12276881393700, 27999904933000, 61578738292500, ... A140903

T(n+7,7) = 1, 120, 4950, 108900, 1557270, 16195608, 131589315, 877262100, 4971151900, 24584605760, 108284013552, 431621592480, 1577078895600, 5337805492800, 16880809870980, 50245234086564, 141622596077325, 379998709805000, 974996689631250, 2401570793407500, ... A140907

T(n+8,8) = 1, 165, 9075, 259545, 4723719, 61408347, 614083470, 4971151900, 33803832920, 198520691512, 1028698128744, 4783805983320, 20239179160200, 78777112731240, 284722993157196, 963033653325810, 3068489581675375, 9262468551496875, ... A140912

T(n+9,9) = 1, 220, 15730, 572572, 13026013, 208416208, 2530768240, 24584605760, 198520691512, 1371597504992, 8291930371088, 44648855844320, 217233856319480, 966332582836544, 3968865965221520, ... A140918

Row sums: 1, 2, 6, 22, 92, 422, 2074, 10754, 58202, 326240, 1882960, 11140560, 67329992, 414499438, 2593341586, ... A1181 Baxter permutations of length n

Triangle 4, based on seed terms:

1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, ...

1 1
1 1 2
1 5 1 7
1 15 15 1 32
1 35 105 35 1 177
1 70 490 490 70 1 1122
1 126 1764 4116 1764 126 1 7898
1 210 5292 24696 24696 5292 210 1 60398
1 330 13860 116424 232848 116424 13860 330 1 494078
1 495 32670 457380 1646568 1646568 457380 32670 495 1 4274228
1 715 70785 1557270 9343620 16818516 9343620 1557270 70785 715 1 38763298
1 1001 143143 4723719 44537922 133613766 133613766 44537922 4723719 143143 1001 1 366039104
1 1365 273273 13026013 184225041 868489479 1447482465 868489479 184225041 13026013 273273 1365 1 3579512809
1 1820 496860 33157124 677352676 4789851066 12544848030 12544848030 4789851066 677352676 33157124 496860 1820 1 36091415154
1 2380 866320 78835120 2254684432 23029990984 90474964580 142174944340 90474964580 23029990984 2254684432 78835120 866320 2380 1 373853631974

Diagonals:

T(n+0,0) = 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A0012

T(n+1,1) = 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, ... A0332

T(n+2,2) = 1, 15, 105, 490, 1764, 5292, 13860, 32670, 70785, 143143, 273273, 496860, 866320, 1456560, 2372112, 3755844, 5799465, 8756055, 12954865, 18818646, ... A6857

T(n+3,3) = 1, 35, 490, 4116, 24696, 116424, 457380, 1557270, 4723719, 13026013, 33157124, 78835120, 176729280, 376375104, 766192176, 1498581756, 2828205765, 5168991135, 9177226366, 15870391460, ... A107915

T(n+4,4) = 1, 70, 1764, 24696, 232848, 1646568, 9343620, 44537922, 184225041, 677352676, 2254684432, 6892441920, 19571505408, 52101067968, 131018862096, 313203587004, 715536058545, 1569305708586, 3316911815140, 6778924352200, ... A47835

T(n+5,5) = 1, 126, 5292, 116424, 1646568, 16818516, 133613766, 868489479, 4789851066, 23029990984, 98561919456, 381644355456, 1354627767168, 4454641311264, 13691471089032, 39620253756006, 108618373687131, 283595960194470, 708397594804900, 1699573176873000, ... A140902

T(n+6,6) = 1, 210, 13860, 457380, 9343620, 133613766, 1447482465, 12544848030, 90474964580, 559299781040, 3031952379456, 14675134144320, 64344818940480, 258616676126160, 962206162645860, 3341308066756506, ... A140904

T(n+7,7) = 1, 330, 32670, 1557270, 44537922, 868489479, 12544848030, 142174944340, 1318349483880, 10323075958624, 69951472754592, 418241323113120, 2241344526426720, ... A140908

T(n+8,8) = 1, 495, 70785, 4723719, 184225041, 4789851066, 90474964580, 1318349483880, 15484613937936, 151561524301616, 1268665346776464, 9271015995674160, ... A140913

T(n+9,9) = 1, 715, 143143, 13026013, 677352676, 23029990984, 559299781040, 10323075958624, 151561524301616, 1832516612010448, ... A140919

Row sums: 1, 2, 7, 32, 177, 1122, 7898, 60398, 494078, 4274228, 38763298, 366039104, 3579512809, 36091415154, 373853631974, ... A5362

Triangle 5, based on seed terms:

1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, ...

1 1
1 1 2
1 6 1 8
1 21 21 1 44
1 56 196 56 1 310
1 126 1176 1176 126 1 2606
1 252 5292 14112 5292 252 1 25202
1 462 19404 116424 116424 19404 462 1 272582
1 792 60984 731808 1646568 731808 60984 792 1 3233738
1 1287 169884 3737448 16818516 16818516 3737448 169884 1287 1 41454272
1 2002 429429 16195608 133613766 267227532 133613766 16195608 429429 2002 1 567709144
1 3003 1002001 61408347 868489479 3184461423 3184461423 868489479 61408347 1002001 3003 1 8230728508
1 4368 2186184 208416208 4789851066 30107635272 55197331332 30107635272 4789851066 208416208 2186184 4368 1 125413517530
1 6188 4504864 644195552 23029990984 235234907908 739309710568 739309710568 235234907908 23029990984 644195552 4504864 6188 1 1996446632130
1 8568 8836464 1837984512 98561919456 1566039386912 7997986868872 13710834632352 7997986868872 1566039386912 98561919456 1837984512 8836464 8568 1 33039704641922

Diagonals:

T(n+0,0) = 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A0012

T(n+1,1) = 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, ... A0389

T(n+2,2) = 1, 21, 196, 1176, 5292, 19404, 60984, 169884, 429429, 1002001, 2186184, 4504864, 8836464, 16604784, 30046752, 52581816, 89311761, 147685461, 238369516, 376372920, ...

T(n+3,3) = 1, 56, 1176, 14112, 116424, 731808, 3737448, 16195608, 61408347, 208416208, 644195552, 1837984512, 4892876352, 12259074816, 29115302688, 65937597264, 143107211709, 298915373064, 603074875480, 1178943365600, ...

T(n+4,4) = 1, 126, 5292, 116424, 1646568, 16818516, 133613766, 868489479, 4789851066, 23029990984, 98561919456, 381644355456, 1354627767168, 4454641311264, 13691471089032, 39620253756006, 108618373687131, 283595960194470, 708397594804900, 1699573176873000, ...

T(n+5,5) = 1, 252, 19404, 731808, 16818516, 267227532, 3184461423, 30107635272, 235234907908, 1566039386912, 9095857138368, 46960429261824, 218772384397632, 931020034054176, 3656383418054268, ...

T(n+6,6) = 1, 462, 60984, 3737448, 133613766, 3184461423, 55197331332, 739309710568, 7997986868872, 72261531710368, 559611782036736, 3792054662892288, ...

T(n+7,7) = 1, 792, 169884, 16195608, 868489479, 30107635272, 739309710568, 13710834632352, 201299981193168, 2424984388825856, ...

T(n+8,8) = 1, 1287, 429429, 61408347, 4789851066, 235234907908, 7997986868872, 201299981193168, 3940599631842016, ...

T(n+9,9) = 1, 2002, 1002001, 208416208, 23029990984, 1566039386912, 72261531710368, 2424984388825856, ...

Row sums: 1, 2, 8, 44, 310, 2606, 25202, 272582, 3233738, 41454272, 567709144, 8230728508, 125413517530, 1996446632130, 33039704641922, ... A5363

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