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A361571 Expansion of e.g.f. exp( (x * (1+x))^3 ).

%I A361571 #15 Jul 06 2025 13:20:27
%S A361571 1,0,0,6,72,360,1080,15120,302400,3689280,32659200,359251200,
%T A361571 6965481600,133880947200,2070484416000,30305353478400,559684629504000,
%U A361571 12582442768896000,271843009108070400,5401042458152140800,111578968350001152000,2657164887872022528000
%N A361571 Expansion of e.g.f. exp( (x * (1+x))^3 ).
%H A361571 Harvey P. Dale, <a href="/A361571/b361571.txt">Table of n, a(n) for n = 0..476</a>
%F A361571 a(n) = n! * Sum_{k=0..floor(n/3)} binomial(3*k,n-3*k)/k!.
%F A361571 a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k * binomial(3,k-3) * a(n-k)/(n-k)!.
%t A361571 With[{nn=30},CoefficientList[Series[Exp[(x(1+x))^3],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jul 06 2025 *)
%o A361571 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x*(1+x))^3)))
%o A361571 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j*binomial(3, j-3)*v[i-j+1]/(i-j)!)); v;
%Y A361571 Cf. A047974, A361570.
%Y A361571 Cf. A361279.
%K A361571 nonn
%O A361571 0,4
%A A361571 _Seiichi Manyama_, Mar 16 2023