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A178657 Irregular triangle: the coefficient [x^k] of the polynomial (1-x)^(2*n-1) * Sum_{s>=0} A001263(n+2*s,2*s+1)*x^s in row n >= 1 and column k >= 0. 1
1, 1, 3, 1, 15, 15, 1, 1, 43, 161, 105, 10, 1, 96, 855, 1680, 855, 96, 1, 1, 185, 3191, 13387, 17655, 7623, 945, 21, 1, 323, 9570, 72254, 188188, 188188, 72254, 9570, 323, 1, 1, 525, 24675, 302359, 1345605, 2499861, 2036125, 715725, 99414, 4410, 36, 1, 808 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row sums are 1, 4, 32, 320, 3584, 43008, 540672, 7028736, 93716480, 1274544128, ... (see A151403, A052704).

The sequence is the Narayana number analog of A034839.

LINKS

Table of n, a(n) for n=1..50.

EXAMPLE

1;

1, 3;

1, 15, 15, 1;

1, 43, 161, 105, 10;

1, 96, 855, 1680, 855, 96, 1;

1, 185, 3191, 13387, 17655, 7623, 945, 21;

1, 323, 9570, 72254, 188188, 188188, 72254, 9570, 323, 1;

1, 525, 24675, 302359, 1345605, 2499861, 2036125, 715725, 99414, 4410, 36;

1, 808, 56896, 1055320, 7329975, 22338816, 32152848, 22338816, 7329975, 1055320, 56896, 808, 1;

MAPLE

A001263 := proc(n, k) if n <=0 or k <=0 then 0 ; elif k > n then 0 ; else binomial(n-1, k-1)*binomial(n, k-1)/k ; end if; end proc:

A178657 := proc(n, k) (1-x)^(2*n-1)*add(A001263(n+2*l, 2*l+1)*x^l, l=0..20) ; expand(%) ; coeftayl(%, x=0, k) ; end proc: # R. J. Mathar, Aug 30 2011

MATHEMATICA

p[x_, n_] = (1 - x)^(2*n - 1)*Sum[(Binomial[2*k + n, 2*k] Binomial[ 2*k + n, 1 + 2*k]/(2*k + n))*x^k, {k, 0, Infinity}];

Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 2, 10}];

Flatten[%]

CROSSREFS

Cf. A001263, A034839.

Sequence in context: A144006 A193966 A113378 * A257490 A156289 A095922

Adjacent sequences:  A178654 A178655 A178656 * A178658 A178659 A178660

KEYWORD

nonn,tabf

AUTHOR

Roger L. Bagula, Jun 01 2010

STATUS

approved

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Last modified February 18 04:48 EST 2020. Contains 332011 sequences. (Running on oeis4.)