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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.

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

EXAMPLE

Row n=0: 1, 1, 1, 1, 1, 1, 1, 1, ... see A000012.

Row n=1: 1, 1, 2, 5, 14, 42, 132, 429, ... see A000108.

Row n=2: 1, 1, 3, 11, 45, 197, 903, 4279, ... see A001003.

Row n=3: 1, 1, 4, 19, 100, 562, 3304, 20071, ... 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

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

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] *)

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.

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 February 20 15:54 EST 2018. Contains 299380 sequences. (Running on oeis4.)