A178619 Triangle T(n,k) with the coefficient of [x^k] of the series (1-x)^(n+1)* sum_{j>=0} binomial(n + 4*j, 4*j)*x^j in row n, column k.

1, 1, 3, 1, 12, 3, 1, 31, 31, 1, 1, 65, 155, 35, 1, 120, 546, 336, 21, 1, 203, 1554, 1918, 413, 7, 1, 322, 3823, 8092, 3823, 322, 1, 1, 486, 8451, 27876, 23607, 4950, 165, 1, 705, 17205, 82885, 112035, 44803, 4455, 55, 1, 990, 32802, 220198, 440484, 291258

Offset

0,3

Comments

Every fourth row is symmetrical.

Row sums are 4^n.

3*k instead of 4*k in the binomial() gives A178618.

Links

Table of n, a(n) for n=0..52.

Example

1;
1, 3;
1, 12, 3;
1, 31, 31, 1;
1, 65, 155, 35;
1, 120, 546, 336, 21;
1, 203, 1554, 1918, 413, 7;
1, 322, 3823, 8092, 3823, 322, 1;
1, 486, 8451, 27876, 23607, 4950, 165;
1, 705, 17205, 82885, 112035, 44803, 4455, 55;
1, 990, 32802, 220198, 440484, 291258, 59950, 2882, 11;

Maple

A178619 := proc(n, k)
    (1-x)^(n+1)*add( binomial(n+4*j, 4*j)*x^j, j=0..n+1) ;
    coeftayl(%, x=0, k) ;
end proc:
seq(seq(A178619(n, k), k=0..n), n=0..8) ; # R. J. Mathar, Nov 05 2012

Mathematica

p[x_, n_] = (-1)^(n + 1)*(-1 + x)^(n + 1)*Sum[Binomial[n + 4*k, 4*k]*x^k, {k, 0, Infinity}]
Flatten[Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]]

Crossrefs

Cf. A178618, A008287.

Sequence in context: A337205 A287197 A118020 * A124572 A144880 A144881

Adjacent sequences: A178616 A178617 A178618 * A178620 A178621 A178622

Keyword

nonn,tabl

Author

Roger L. Bagula, May 30 2010

Status

approved