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A249941 E.g.f.: Sum_{n>=0} exp(n^3*x) / 2^(n+1).

%I A249941 #33 Jan 17 2022 07:07:40
%S A249941 1,13,4683,7087261,28091567595,230283190977853,3385534663256845323,
%T A249941 81124824998504073881821,2958279121074145472650648875,
%U A249941 155897763918621623249276226253693,11403568794011880483742464196184901963,1120959742203056268267494209293006882589981
%N A249941 E.g.f.: Sum_{n>=0} exp(n^3*x) / 2^(n+1).
%C A249941 Number of ordered partitions of 3*n.
%H A249941 Seiichi Manyama, <a href="/A249941/b249941.txt">Table of n, a(n) for n = 0..141</a>
%F A249941 a(n) = Sum_{k=0..3*n} k! * Stirling2(3*n, k) for n>=0.
%F A249941 a(n) = Sum_{k=1..[(3*n+1)/2]} (2*k-1)! * Stirling2(3*n+1, 2*k) for n>0 with a(0)=1.
%F A249941 a(n) = A000670(3*n), where A000670 is the Fubini numbers.
%F A249941 a(n) ~ (3*n)! / (2 * (log(2))^(3*n+1)). - _Vaclav Kotesovec_, May 04 2015
%F A249941 a(n) = Sum_{k>=0} k^(3*n) / 2^(k + 1). - _Ilya Gutkovskiy_, Dec 19 2019
%e A249941 E.g.f.: A(x) = 1 + 13*x + 4683*x^2/2! + 7087261*x^3/3! + 28091567595*x^4/4! +...
%e A249941 where the e.g.f. equals the infinite series:
%e A249941 A(x) = 1/2 + exp(x)/2^2 + exp(8*x)/2^3 + exp(27*x)/2^4 + exp(64*x)/2^5 + exp(125*x)/2^6 + exp(216*x)/2^7 + exp(343*x)/2^8 +...
%t A249941 Table[Sum[k! * StirlingS2[3*n, k],{k,0,3*n}],{n,0,20}] (* _Vaclav Kotesovec_, May 04 2015 *)
%t A249941 Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; a[n_] := Fubini[3n, 1]; a[0] = 1; Table[a[n], {n, 0, 11}] (* _Jean-François Alcover_, Mar 30 2016 *)
%o A249941 (PARI) /* E.g.f.: Sum_{n>=0} exp(n^3*x)/2^(n+1) */
%o A249941 \p100 \\ set precision
%o A249941 {a(n) = round( n!*polcoeff(sum(m=0,600,exp(m^3*x +x*O(x^n))/2^(m+1)*1.),n) )}
%o A249941 for(n=0,20,print1(a(n),", "))
%o A249941 (PARI) /* Formula for a(n): */
%o A249941 {a(n) = sum(k=0, 3*n, k! * stirling(3*n, k, 2))}
%o A249941 for(n=0, 20, print1(a(n), ", "))
%o A249941 (PARI) /* Formula for a(n): */
%o A249941 {a(n) = if(n==0,1, sum(k=1,(3*n+1)\2, (2*k-1)! * stirling(3*n+1, 2*k, 2)))}
%o A249941 for(n=0,20,print1(a(n),", "))
%Y A249941 Cf. A000670, A068942, A249940.
%K A249941 nonn
%O A249941 0,2
%A A249941 _Paul D. Hanna_, Nov 19 2014