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, 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.

Links

G. C. Greubel, Rows n=0..100 of triangle, flattened

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 A377876 A206493

Adjacent sequences: A178664 A178665 A178666 * A178668 A178669 A178670

Keyword

nonn,tabf

Author

Roger L. Bagula, Jun 02 2010

Status

approved