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%I #7 Nov 06 2012 03:37:26
%S 1,1,3,1,12,3,1,31,31,1,1,65,155,35,1,120,546,336,21,1,203,1554,1918,
%T 413,7,1,322,3823,8092,3823,322,1,1,486,8451,27876,23607,4950,165,1,
%U 705,17205,82885,112035,44803,4455,55,1,990,32802,220198,440484,291258
%N 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.
%C Every fourth row is symmetrical.
%C Row sums are 4^n.
%C 3*k instead of 4*k in the binomial() gives A178618.
%e 1;
%e 1, 3;
%e 1, 12, 3;
%e 1, 31, 31, 1;
%e 1, 65, 155, 35;
%e 1, 120, 546, 336, 21;
%e 1, 203, 1554, 1918, 413, 7;
%e 1, 322, 3823, 8092, 3823, 322, 1;
%e 1, 486, 8451, 27876, 23607, 4950, 165;
%e 1, 705, 17205, 82885, 112035, 44803, 4455, 55;
%e 1, 990, 32802, 220198, 440484, 291258, 59950, 2882, 11;
%p A178619 := proc(n,k)
%p (1-x)^(n+1)*add( binomial(n+4*j,4*j)*x^j,j=0..n+1) ;
%p coeftayl(%,x=0,k) ;
%p end proc:
%p seq(seq(A178619(n,k),k=0..n),n=0..8) ; # _R. J. Mathar_, Nov 05 2012
%t p[x_, n_] = (-1)^(n + 1)*(-1 + x)^(n + 1)*Sum[Binomial[n + 4*k, 4*k]*x^k, {k, 0, Infinity}]
%t Flatten[Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]]
%Y Cf. A178618, A008287.
%K nonn,tabl
%O 0,3
%A _Roger L. Bagula_, May 30 2010
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