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A243942 O.g.f.: Sum_{n>=0} n^(2*n) * x^n / (1 - n*x)^n * exp( -n^2*x / (1 - n*x) ) / n!.

%I A243942 #22 Aug 11 2014 09:52:50
%S A243942 1,1,8,121,2698,79654,2929238,129004633,6619919386,387904397222,
%T A243942 25555935470016,1869945551975658,150459006927310348,
%U A243942 13203459856456213172,1254972882696473807298,128439184335788533011489,14082139161229781077548346,1646731810035799151750487814
%N A243942 O.g.f.: Sum_{n>=0} n^(2*n) * x^n / (1 - n*x)^n * exp( -n^2*x / (1 - n*x) ) / n!.
%F A243942 a(n) = Sum_{k=1..n} C(n-1,k-1) * Stirling2(n+k,k) for n>0, a(0)=1.
%F A243942 a(n) = c * (r^2/((1-r)*(2*r-1)))^n * n^(n-1/2) / exp(n), where r = 0.859294411517830517100430385442711799997876163... is the root of the equation (1-r)*(1+r)/r^2 = -LambertW(-exp(-1-1/r)*(1+r)/r), and c = 0.4180257159270405799046057130547446708890452... . - _Vaclav Kotesovec_, Aug 11 2014
%e A243942 O.g.f.: A(x) = 1 + x + 8*x^2 + 121*x^3 + 2698*x^4 + 79654*x^5 + 2929238*x^6 +...
%e A243942 where
%e A243942 A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^4*x^2/(1-2*x)^2*exp(-4*x/(1-2*x))/2! + 3^6*x^3/(1-3*x)^3*exp(-9*x/(1-3*x))/3! + 4^8*x^4/(1-4*x)^4*exp(-16*x/(1-4*x))/4! + 5^10*x^5/(1-5*x)^5*exp(-25*x/(1-5*x))/5! +...
%e A243942 simplifies to a power series in x with integer coefficients.
%e A243942 Illustrate the terms by:
%e A243942 a(1) = 1*1 = 1;
%e A243942 a(2) = 1*1 + 1*7 = 8;
%e A243942 a(3) = 1*1 + 2*15 + 1*90 = 121;
%e A243942 a(4) = 1*1 + 3*31 + 3*301 + 1*1701 = 2698;
%e A243942 a(5) = 1*1 + 4*63 + 6*966 + 4*7770 + 1*42525 = 79654; ...
%e A243942 where Stirling2(n+k,k) forms a rectangular table as follows:
%e A243942 1, 1,  1,   1,    1,     1,      1,      1, ...;
%e A243942 0, 1,  3,   6,   10,    15,     21,     28, ...;
%e A243942 0, 1,  7,  25,   65,   140,    266,    462, ...;
%e A243942 0, 1, 15,  90,  350,  1050,   2646,   5880, ...;
%e A243942 0, 1, 31, 301, 1701,  6951,  22827,  63987, ...;
%e A243942 0, 1, 63, 966, 7770, 42525, 179487, 627396, ...; ...
%t A243942 Flatten[{1,Table[Sum[Binomial[n-1,k-1] * StirlingS2[n+k,k],{k,1,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Aug 11 2014 *)
%o A243942 (PARI) {a(n)=polcoeff(sum(k=0, n+1, (k^2*x)^k/(1-k*x)^k*exp(-k^2*x/(1-k*x+x*O(x^n)))/k!), n)}
%o A243942 for(n=0, 25, print1(a(n), ", "))
%o A243942 (PARI) {Stirling2(n, k) = sum(j=0, k, (-1)^(k+j) * binomial(k, j) * j^n) / k!}
%o A243942 {a(n)=if(n==0, 1, sum(k=1, n, Stirling2(n+k, k) * binomial(n-1, k-1)))}
%o A243942 for(n=0, 30, print1(a(n), ", "))
%Y A243942 Cf. A134055, A174845, A218667, A048993 (Stirling2).
%K A243942 nonn
%O A243942 0,3
%A A243942 _Paul D. Hanna_, Aug 09 2014