A178618 Triangle T(n,k) with the coefficient [x^k] of the series (1-x)^(n+1) * Sum_{j>=0} 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

0,3

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.

Links

Table of n, a(n) for n=0..50.

Formula

T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(n+1,k-j) * binomial(n+3*j,3*j). - Seiichi Manyama, Dec 24 2025

Example

Triangular array begins:
  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;

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}]]

Prog

(PARI) T(n, k) = sum(j=0, k, (-1)^(k-j)*binomial(n+1, k-j)*binomial(n+3*j, 3*j)); \\ Seiichi Manyama, Dec 24 2025

Crossrefs

Cf. A034839, A178619,

Cf. A094531, A111808, A027907, A086645.

Sequence in context: A116891 A079620 A010254 * A155992 A178211 A305564

Adjacent sequences: A178615 A178616 A178617 * A178619 A178620 A178621

Keyword

nonn,tabf

Author

Roger L. Bagula, May 30 2010

Extensions

Keyword fixed by Seiichi Manyama, Dec 24 2025

Status

approved