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%I #10 Dec 09 2015 18:36:50
%S 1,2,6,299,2630475,5194717544512,3708580324835714831258,
%T 1461034854533485247412937306733067,
%U 450538787986948219326155652866541933427447505845,146413934927214452212855330835382443952802497537220927026261910086,64954656894649578286072291957497800888821547620744164639686415856667115998603764278502
%N G.f.: Sum_{n>=0} (n^2)!/n!^(n+1) * x^n / (1-x)^(n^2+1).
%H Paul D. Hanna, <a href="/A264410/b264410.txt">Table of n, a(n) for n = 0..30</a>
%F a(n) = Sum_{k=0..n} (1/k!) * Product_{j=0..k-1} binomial(n+j*k, k).
%F a(n) ~ exp(n - 1/12) * n^(n^2 - 3*n/2 + 1/2) / (2*Pi)^(n/2). - _Vaclav Kotesovec_, Dec 09 2015
%e G.f.: A(x) = 1 + 2*x + 9*x^2 + 1714*x^3 + 63079895*x^4 + 623361815288736*x^5 +...
%e where
%e A(x) = 1/(1-x) + x/(1-x)^2 + (4!/2!^3)*x^2/(1-x)^5 + (9!/3!^4)*x^3/(1-x)^10 + (16!/4!^5)*x^4/(1-x)^17 + (25!/5!^6)*x^5/(1-x)^26 +...
%e Equivalently,
%e A(x) = 1/(1-x) + x/(1-x)^2 + 3*x^2/(1-x)^5 + 280*x^3/(1-x)^10 + 2627625*x^4/(1-x)^17 + 5194672859376*x^5/(1-x)^26 +...+ A057599(n)*x^n/(1-x)^(n^2+1) +...
%e Illustrate formula a(n) = Sum_{k=0..n} (1/k!)*Product_{j=0..k-1} C(n+j*k,k) for initial terms:
%e a(0) = 1;
%e a(1) = 1 + C(1,1);
%e a(2) = 1 + C(2,1) + C(2,2)*C(4,2)/2!;
%e a(3) = 1 + C(3,1) + C(3,2)*C(5,2)/2! + C(3,3)*C(6,3)*C(9,3)/3!;
%e a(4) = 1 + C(4,1) + C(4,2)*C(6,2)/2! + C(4,3)*C(7,3)*C(10,3)/3! + C(4,4)*C(8,4)*C(12,4)*C(16,4)/4!;
%e a(5) = 1 + C(5,1) + C(5,2)*C(7,2)/2! + C(5,3)*C(8,3)*C(11,3)/3! + C(5,4)*C(9,4)*C(13,4)*C(17,4)/4! + C(5,5)*C(10,5)*C(15,5)*C(20,5)*C(25,5)/5!; ...
%e which numerically equals:
%e a(0) = 1;
%e a(1) = 1 + 1 = 2;
%e a(2) = 1 + 2 + 1*6/2! = 6;
%e a(3) = 1 + 3 + 3*10/2! + 1*20*84/3! = 299;
%e a(4) = 1 + 4 + 6*15/2! + 4*35*120/3! + 1*70*495*1820/4! = 2630475;
%e a(5) = 1 + 5 + 10*21/2! + 10*56*165/3! + 5*126*715*2380/4! + 1*252*3003*15504*53130/5! = 5194717544512; ...
%t Table[Sum[Product[Binomial[n+j*k, k], {j, 0, k-1}]/k!, {k, 0, n}], {n, 0, 10}] (* _Vaclav Kotesovec_, Dec 09 2015 *)
%o (PARI) {a(n)=polcoeff(sum(m=0, n, (m^2)!/m!^(m+1)*x^m/(1-x +x*O(x^n))^(m^2+1)), n)}
%o for(n=0, 15, print1(a(n), ", "))
%o (PARI) {a(n) = sum(k=0, n, (1/k!) * prod(j=0, k-1, binomial(n+j*k, k)) )}
%o for(n=0, 15, print1(a(n), ", "))
%Y Cf. A229050, A057599.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 26 2015
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