%I #10 Jan 20 2018 17:32:21
%S 1,3,177,43743,28317777,37918359903,91064083658577,356470099797125343,
%T 2123580647871774583377,18282562085069810089566303,
%U 218479480936045179472923760977,3508620018746019243855156135806943,73737548542861221762649623289597264977
%N G.f.: Sum_{n>=0} (3*n)!/2^n * x^n / Product_{k=1..n} (1 + k^3*x).
%C Compare to the o.g.f. for Euler numbers (A000364):
%C Sum_{n>=0} (2*n)!/2^n * x^n / Product_{k=1..n} (1 + k^2*x).
%H Paul D. Hanna, <a href="/A216967/b216967.txt">Table of n, a(n) for n = 0..100</a>
%H L. Naughton, G. Pfeiffer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Naughton/naughton2.html">Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group</a>, J. Int. Seq. 16 (2013) #13.5.8
%F a(n) == 0 (mod 3) for n>0;
%F a(n) == 3 (mod 6) for n>0;
%F a(2*n-1) == 3 (mod 5), a(2*n) == 2 (mod 5), for n>0;
%F a(2*n-1) == 3 (mod 9), a(2*n) == 6 (mod 9), for n>0;
%F a(2*n-1) == 3 (mod 10), a(2*n) == 7 (mod 10), for n>0.
%F a(n) ~ c * 2^(3/2) * Pi^(3/2) * d^n * n^(3*n+1/2) / exp(3*n), where d = 13.2458829063958687527098..., c = 0.281041890716214414121... . - _Vaclav Kotesovec_, Dec 05 2015
%e G.f.: A(x) = 1 + 3*x + 177*x^2 + 43743*x^3 + 28317777*x^4 +...
%o (PARI) {a(n)=polcoeff(sum(m=0,n,(3*m)!/2^m*x^m/prod(k=1,m,1+k^3*x+x*O(x^n))),n)}
%o for(n=0,21,print1(a(n),", "))
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 20 2012