cp's OEIS Frontend

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.

%I A178618 #7 Jan 03 2024 07:38:40
%S A178618 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,
%T A178618 784,1016,266,8,1,156,1554,3139,1554,156,1,1,210,2850,8350,6765,1452,
%U A178618 55,1,275,4917,19855,24068,9042,880,11
%N 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.
%C A178618 Every third row is symmetrical.
%C A178618 Row sums are 3^n.
%C A178618 2*k instead of 3*k in the binomial() gives A034839 with alternating rows of A086645.
%e A178618 1;
%e A178618 1, 2;
%e A178618 1, 7, 1;
%e A178618 1, 16, 10;
%e A178618 1, 30, 45, 5;
%e A178618 1, 50, 141, 50, 1;
%e A178618 1, 77, 357, 266, 28;
%e A178618 1, 112, 784, 1016, 266, 8;
%e A178618 1, 156, 1554, 3139, 1554, 156, 1;
%e A178618 1, 210, 2850, 8350, 6765, 1452, 55;
%e A178618 1, 275, 4917, 19855, 24068, 9042, 880, 11;
%p A178618 A178618 := proc(n,k)
%p A178618     (1-x)^(n+1)*add( binomial(n+3*j,3*j)*x^j,j=0..n+1) ;
%p A178618     coeftayl(%,x=0,k) ;
%p A178618 end proc:
%p A178618 seq(seq(A178618(n,k),k=0..n),n=0..8) ; # _R. J. Mathar_, Nov 05 2012
%t A178618 p[x_, n_] = (-1)^(n + 1)*(-1 + x)^(n + 1)*Sum[Binomial[n + 3*k, 3*k]*x^k, {k, 0, Infinity}]
%t A178618 Flatten[Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]]
%Y A178618 Cf. A094531, A111808, A027907.
%K A178618 nonn,tabl
%O A178618 0,3
%A A178618 _Roger L. Bagula_, May 30 2010