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

Author

Roger L. Bagula, May 30 2010

Keywords

• nonn
• tabl

Comments

Every fourth row is symmetrical.

Row sums are 4^n.

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

Examples

			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;
		

Crossrefs

Cf. A178618, A008287.

Programs

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}]]