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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
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.
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
Sequence in context: A366120 A113378 A365162 * A257490 A156289 A368493
KEYWORD
nonn,tabf
AUTHOR
Roger L. Bagula, Jun 01 2010
STATUS
approved