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A178667
Irregular triangle: T(n,k) is the coefficient [x^k] of the series (-1)^n *(x-1)^(n+2) *sum_{j=0..infinity} x^j /Beta(n+1,2*j+1), k=0..1+n/2, where Beta() is the usual Gamma-function ratio.
1
1, 1, 2, 6, 3, 18, 3, 4, 40, 20, 5, 75, 75, 5, 6, 126, 210, 42, 7, 196, 490, 196, 7, 8, 288, 1008, 672, 72, 9, 405, 1890, 1890, 405, 9, 10, 550, 3300, 4620, 1650, 110, 11, 726, 5445, 10164, 5445, 726, 11
OFFSET
0,3
COMMENTS
The even-indexed rows (at least if limited to k<=1+n/2) are left-right symmetric.
EXAMPLE
1, 1;
2, 6;
3, 18, 3;
4, 40, 20;
5, 75, 75, 5;
6, 126, 210, 42;
7, 196, 490, 196, 7;
8, 288, 1008, 672, 72;
9, 405, 1890, 1890, 405, 9;
10, 550, 3300, 4620, 1650, 110;
11, 726, 5445, 10164, 5445, 726, 11;
MAPLE
A178667 := proc(n, k)
(-1)^n*(x-1)^(n+2)*add(x^j/Beta(n+1, 2*j+1), j=0..n+1) ;
coeftayl(%, x=0, k) ;
end proc: # R. J. Mathar, Feb 12 2013
MATHEMATICA
p[x_, n_] = (-1)^n*(-1 + x)^(n + 2)*Sum[(1/Beta[n + 1, 2*k + 1])x^k, {k, 0, Infinity}];
Flatten[Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]]
CROSSREFS
Cf. A036289 (Row sums).
Sequence in context: A083169 A276817 A050125 * A281881 A206493 A359256
KEYWORD
nonn,tabf
AUTHOR
Roger L. Bagula, Jun 02 2010
STATUS
approved