A008550 Table T(n,k), n>=0 and k>=0, read by antidiagonals: the k-th column given by the k-th Narayana polynomial.

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

Offset: 0

Author

Philippe Deléham, Jan 23 2004

Keywords

• easy
• nonn
• tabl

Comments

Mirror image of A243631. - Philippe Deléham, Sep 26 2014

Examples

			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.
... - _Philippe Deléham_, Apr 03 2013
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;
  1, 42, 45, 19, 5, 1, 1; - _G. C. Greubel_, Feb 15 2021
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475
• H. Prodinger, On a functional difference equation of Runyon, Morrison, Carlitz, and Riordan, arXiv:math/0103149 [math.CO], 2001.
• H. Prodinger, On a functional difference equation of Runyon, Morrison, Carlitz, and Riordan, Séminaire Lotharingien de Combinatoire 46 (2001), Article B46a.
• L. Yang, S.-L. Yang, A relation between Schroder paths and Motzkin paths, Graphs Combinat. 36 (2020) 1489-1502, eq. (6).

Crossrefs

Columns: A000012, A000012, A000027, A028387, A090197, A090198, A090199, A090200.

Main diagonal is A242369.

A diagonal is in A099169.

Cf. A204057 (another version), A088617, A243631.

Cf. A132745.

Programs

Magma

[Truncate(HypergeometricSeries(k-n, k-n+1, 2, k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 15 2021

Maple

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

Mathematica

(* 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 *)

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

Formula

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