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A008550 Table T(n,k), n>=0 and k>=0, read by antidiagonals: the k-th column given by the k-th Narayana polynomial. 9
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

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

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).

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

EXAMPLE

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

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 *)

PROG

(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

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.

Sequence in context: A155586 A069739 A066060 * A064094 A090182 A256384

Adjacent sequences:  A008547 A008548 A008549 * A008551 A008552 A008553

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Jan 23 2004

STATUS

approved

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Last modified May 16 11:35 EDT 2022. Contains 353704 sequences. (Running on oeis4.)