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%I #12 Sep 25 2013 19:50:04
%S 1,3,29,13567,1009142769,19947560933879891,
%T 170891375663413844489045533,942542805274443250197129297402029958879,
%U 4650425326497533656923054675764068523027405525255181377,27202912617670436035808496756146798219927348043651854025145948950565355779
%N G.f.: Sum_{n>=0} (n^2)!/n!^n * (2*x)^n / (1-x)^(n^2+1).
%H Alois P. Heinz, <a href="/A229051/b229051.txt">Table of n, a(n) for n = 0..26</a>
%F a(n) = Sum_{k=0..n} 2^k * Product_{j=0..k-1} binomial(n+j*k,k).
%F a(n) ~ exp(-1/12) * n^(n^2-n/2+1) * 2^n / (2*Pi)^((n-1)/2). - _Vaclav Kotesovec_, Sep 23 2013
%e G.f.: A(x) = 1 + 3*x + 29*x^2 + 13567*x^3 + 1009142769*x^4 +...
%e where
%e A(x) = 1/(1-x) + (2*x)/(1-x)^2 + (4!/2!^2)*(2*x)^2/(1-x)^5 + (9!/3!^3)*(2*x)^3/(1-x)^10 + (16!/4!^4)*(2*x)^4/(1-x)^17 + (25!/5!^5)*(2*x)^5/(1-x)^26 +...
%e Equivalently,
%e A(x) = 1/(1-x) + (2*x)/(1-x)^2 + 6*(2*x)^2/(1-x)^5 + 1680*(2*x)^3/(1-x)^10 + 63063000*(2*x)^4/(1-x)^17 + 623360743125120*(2*x)^5/(1-x)^26 +...+ A034841(n)*(2*x)^n/(1-x)^(n^2+1) +...
%e Illustrate formula a(n) = Sum_{k=0..n} 2^k * Product_{j=0..k-1} C(n+j*k,k) for initial terms:
%e a(0) = 1;
%e a(1) = 1 + 2*C(1,1);
%e a(2) = 1 + 2*C(2,1) + 4*C(2,2)*C(4,2);
%e a(3) = 1 + 2*C(3,1) + 4*C(3,2)*C(5,2) + 8*C(3,3)*C(6,3)*C(9,3);
%e a(4) = 1 + 2*C(4,1) + 4*C(4,2)*C(6,2) + 8*C(4,3)*C(7,3)*C(10,3) + 16*C(4,4)*C(8,4)*C(12,4)*C(16,4);
%e a(5) = 1 + 2*C(5,1) + 4*C(5,2)*C(7,2) + 8*C(5,3)*C(8,3)*C(11,3) + 16*C(5,4)*C(9,4)*C(13,4)*C(17,4) + 32*C(5,5)*C(10,5)*C(15,5)*C(20,5)*C(25,5); ...
%e which numerically equals:
%e a(0) = 1;
%e a(1) = 1 + 2*1 = 3;
%e a(2) = 1 + 2*2 + 4*1*6 = 29;
%e a(3) = 1 + 2*3 + 4*3*10 + 8*1*20*84 = 13567;
%e a(4) = 1 + 2*4 + 4*6*15 + 8*4*35*120 + 16*1*70*495*1820 = 1009142769;
%e a(5) = 1 + 2*5 + 4*10*21 + 8*10*56*165 + 16*5*126*715*2380 + 32*1*252*3003*15504*53130 = 19947560933879891; ...
%p with(combinat):
%p a:= n-> add(2^k*multinomial(n+(k-1)*k, n-k, k$k), k=0..n):
%p seq(a(n), n=0..10); # _Alois P. Heinz_, Sep 23 2013
%t Table[Sum[2^k*Product[Binomial[n+j*k,k],{j,0,k-1}],{k,0,n}],{n,0,10}] (* _Vaclav Kotesovec_, Sep 23 2013 *)
%o (PARI) {a(n)=polcoeff(sum(m=0, n, (m^2)!/m!^m*(2*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,2^k*prod(j=0,k-1,binomial(n+j*k,k)))}
%o for(n=0,15,print1(a(n),", "))
%Y Cf. A229050, A034841; A001850, A081798, A082488, A082489, A229049.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 22 2013
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