A187667 Coefficient of x^n in (1 + n*x + n*x^2 + n*x^3)^n.
1, 1, 8, 90, 1312, 23625, 505116, 12475596, 348942336, 10888165395, 374606200000, 14077548113398, 573396296212224, 25150850370412156, 1181513742628738624, 59165118490203450000, 3145241884988171878400, 176865209305943158023799, 10486960289673977419520256
Offset: 0
Programs
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Maple
A027907 := proc(n,k) add( binomial(n,j)*binomial(n-j,k-2*j),j=0..n) ; end proc: A187667 := proc(n) add( binomial(n,k)*A027907(k,n-k)*n^k,k=0..n) ; end proc: seq(A187667(n),n=0..10) ; # R. J. Mathar, Mar 27 2011
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Mathematica
a[n_] := If[n == 0, 1, Coefficient[(1 + n x + n x^2 + n x^3)^n, x^n]] Table[a[n], {n, 0, 12}] -
Maxima
makelist(coeff(expand((1+n*x+n*x^2+n*x^3)^n),x,n),n,0,12);
Formula
a(n) = Sum_{k=0..n} binomial(n,k)*trinomial(k,n-k)*n^k.