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A008550
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Table T(n,k), n>=0 and k>=0, read by antidiagonals: the k-th column given by the k-th Narayana polynomial.
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9
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 11, 4, 1, 1, 1, 42, 45, 19, 5, 1, 1, 1, 132, 197, 100, 29, 6, 1, 1, 1, 429, 903, 562, 185, 41, 7, 1, 1, 1, 1430, 4279, 3304, 1257, 306, 55, 8, 1, 1, 1, 4862, 20793, 20071, 8925, 2426, 469, 71, 9, 1, 1
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OFFSET
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0,8
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COMMENTS
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LINKS
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FORMULA
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T(n, k) = Sum_{j>0} A001263(k, j)*n^(j-1); T(n, 0)=1.
T(n, k) = Sum_{j, 0<=j<=k} A088617(k, j)*n^j*(1-n)^(k-j).
The o.g.f. of row n is gf(n) = 2/(sqrt((n-1)^2*x^2-2*(n+1)*x+1)+(n-1)*x+1). - Peter Luschny, Nov 17 2014
G.f. of row n: 1/(1 - x/(1 - n*x/(1 - x/(1 - n*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Aug 10 2017
T(n, k) = Hypergeometric2F1([k-n, k-n+1], [2], k), as a number triangle. - G. C. Greubel, Feb 15 2021
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EXAMPLE
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Row n=0: 1, 1, 1, 1, 1, 1, 1, ... see A000012.
Row n=1: 1, 1, 2, 5, 14, 42, 132, ... see A000108.
Row n=2: 1, 1, 3, 11, 45, 197, 903, ... see A001003.
Row n=3: 1, 1, 4, 19, 100, 562, 3304, ... see A007564.
Row n=4: 1, 1, 5, 29, 185, 1257, 8925, ... see A059231.
Row n=5: 1, 1, 6, 41, 306, 2426, 20076, ... see A078009.
Row n=6: 1, 1, 7, 55, 469, 4237, 39907, ... see A078018.
Row n=7: 1, 1, 8, 71, 680, 6882, 72528, ... see A081178.
Row n=8: 1, 1, 9, 89, 945, 10577, 123129, ... see A082147.
Row n=9: 1, 1, 10, 109, 1270, 15562, 198100, ... see A082181.
Row n=10: 1, 1, 11, 131, 161, 1661, 22101, ... see A082148.
Row n=11: 1, 1, 12, 155, 2124, 30482, 453432, ... see A082173.
The first few rows of the antidiagonal triangle are:
1;
1, 1;
1, 1, 1;
1, 2, 1, 1;
1, 5, 3, 1, 1;
1, 14, 11, 4, 1, 1;
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MAPLE
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gf := n -> 2/(sqrt((n-1)^2*x^2-2*(n+1)*x+1)+(n-1)*x+1):
for n from 0 to 11 do PolynomialTools:-CoefficientList(convert( series(gf(n), x, 12), polynom), x) od; # Peter Luschny, Nov 17 2014
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MATHEMATICA
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(* First program *)
Unprotect[Power]; Power[0 | 0, 0 | 0] = 1; Protect[Power]; Table[Function[n, Sum[Apply[Binomial[#1 + #2, #1] Binomial[#1, #2]/(#2 + 1) &, {k, j}]*n^j*(1 - n)^(k - j), {j, 0, k}]][m - k + 1] /. k_ /; k <= 0 -> 1, {m, -1, 9}, {k, m + 1, 0, -1}] // Flatten (* Michael De Vlieger, Aug 10 2017 Note: this code renders 0^0 = 1. To restore normal Power functionality: Unprotect[Power]; ClearAll[Power]; Protect[Power] *)
(* Second program *)
Table[Hypergeometric2F1[1-n+k, k-n, 2, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 15 2021 *)
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PROG
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(Sage) flatten([[hypergeometric([k-n, k-n+1], [2], k).simplify_hypergeometric() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 15 2021
(Magma) [Truncate(HypergeometricSeries(k-n, k-n+1, 2, k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 15 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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