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%I #15 Jan 21 2020 18:26:12
%S 1,1,11,365,25323,3068521,583027547,161601254725,62042488237755,
%T 31728742163212641,20963751508027371691,17461136553331587079965,
%U 17967906090023681913528523,22459900935806853610377326041,33617974358392980795259947648187,59515082206147526028817472280664565
%N a(n) = (3*n)! * [z^(3*n)] exp(exp(z)/3 + 2*exp(-z/2)*cos(z*sqrt(3)/2)/3 - 1).
%C Row sums of A291451.
%H Andrew Howroyd, <a href="/A291973/b291973.txt">Table of n, a(n) for n = 0..100</a>
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(3*n-1,3*k-1) * a(n-k). - _Ilya Gutkovskiy_, Jan 21 2020
%p A291973 := proc(n) exp(exp(z)/3+2*exp(-z/2)*cos(z*sqrt(3)/2)/3-1):
%p (3*n)!*coeff(series(%, z, 3*(n+1)), z, 3*n) end:
%p seq(A291973(n), n=0..15);
%t P[m_, n_] := P[m, n] = If[n == 0, 1, Sum[Binomial[m*n, m*k]*P[m, n - k]*x, {k, 1, n}]];
%t a[n_] := Module[{cl = CoefficientList[P[3, n], x]}, Sum[cl[[k + 1]]/k!, {k, 0, n}]];
%t Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Jul 23 2019, after _Peter Luschny_ in A291451 *)
%o (PARI) seq(n)={my(a=vector(n+1)); a[1]=1; for(n=1, n, a[1+n]=sum(k=1, n, binomial(3*n-1, 3*k-1) * a[1+n-k])); a} \\ _Andrew Howroyd_, Jan 21 2020
%Y Cf. A291451.
%K nonn
%O 0,3
%A _Peter Luschny_, Sep 07 2017
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