A178618 Triangle T(n,k) with the coefficient [x^k] of the series (1-x)^(n+1) * sum_{j=0..infinity} *binomial(n+3*j,3*j)*x^j, in row n, column k.

1, 1, 2, 1, 7, 1, 1, 16, 10, 1, 30, 45, 5, 1, 50, 141, 50, 1, 1, 77, 357, 266, 28, 1, 112, 784, 1016, 266, 8, 1, 156, 1554, 3139, 1554, 156, 1, 1, 210, 2850, 8350, 6765, 1452, 55, 1, 275, 4917, 19855, 24068, 9042, 880, 11

Offset: 0

Author

Roger L. Bagula, May 30 2010

Keywords

• nonn
• tabl

Comments

Every third row is symmetrical.

Row sums are 3^n.

2*k instead of 3*k in the binomial() gives A034839 with alternating rows of A086645.

Examples

			1;
1, 2;
1, 7, 1;
1, 16, 10;
1, 30, 45, 5;
1, 50, 141, 50, 1;
1, 77, 357, 266, 28;
1, 112, 784, 1016, 266, 8;
1, 156, 1554, 3139, 1554, 156, 1;
1, 210, 2850, 8350, 6765, 1452, 55;
1, 275, 4917, 19855, 24068, 9042, 880, 11;
		

Crossrefs

Cf. A094531, A111808, A027907.

Programs

Maple

A178618 := proc(n,k)
    (1-x)^(n+1)*add( binomial(n+3*j,3*j)*x^j,j=0..n+1) ;
    coeftayl(%,x=0,k) ;
end proc:
seq(seq(A178618(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 + 3*k, 3*k]*x^k, {k, 0, Infinity}]
Flatten[Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]]