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%I #13 May 14 2014 16:47:17
%S 1,3124,191757120,49208861869440,33030777426968816640,
%T 45829974166034718596428800,114009204539207742166715857223680,
%U 462192193445890293982679086838571270144,2851153321165202191241172917762717987236478976
%N O.g.f.: Sum_{n>=0} (4*n+1)^(4*n+1) * exp(-(4*n+1)^4*x) * x^n / n!.
%C From _Vaclav Kotesovec_, May 13 2014: (Start)
%C Generally, for p>1, a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (p*k+1)^(p*n+1) = Sum_{k=0..(p-1)*n+1} p^(n+k) * binomial(p*n+1,n+k) * stirling2(n+k,n).
%C a(n) ~ n^(n*p-n+1/2) * p^(2*p*n+2+1/p) / (sqrt(2*Pi*(1-r)) * exp((p-1)*n) * r^(n+1/p) * (p-r)^(n*p-n+1)), where r = -LambertW(-p*exp(-p)).
%C (End)
%F a(n) = 1/n! * [x^n] Sum_{k>=0} (4*k+1)^(4*k+1) * x^k / (1 + (4*k+1)^4*x)^(k+1).
%F a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (4*k+1)^(4*n+1).
%F a(n) ~ n^(3*n+1/2) * 2^(16*n+9/2) / (sqrt(2*Pi*(1-r)) * exp(3*n) * r^(n+1/4) * (4-r)^(3*n+1)), where r = -LambertW(-4*exp(-4)) = 0.0793096051271136564391... . - _Vaclav Kotesovec_, May 13 2014
%e O.g.f.: A(x) = 1 + 3124*x + 191757120*x^2 + 49208861869440*x^3 +...
%e where
%e A(x) = exp(-x) + 5^5*x*exp(-5^4*x) + 9^9*exp(-9^4*x)*x^2/2! + 13^13*exp(-13^4*x)*x^3/3! + 17^17*exp(-17^4*x)*x^4/4! + 21^21*exp(-21^4*x)*x^5/5! +...
%e is a power series in x with integer coefficients.
%t Table[1/n!*Sum[(-1)^(n-k)*Binomial[n,k]*(4*k+1)^(4*n+1),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, May 13 2014 *)
%t Table[Sum[Binomial[4*n+1,n+k]*4^(n+k)*StirlingS2[n+k,n],{k,0,3*n+1}],{n,0,20}] (* _Vaclav Kotesovec_, May 13 2014 *)
%o (PARI) {a(n)=polcoeff(sum(k=0, n, (4*k+1)^(4*k+1)*exp(-(4*k+1)^4*x +x*O(x^n))*x^k/k!), n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=(1/n!)*polcoeff(sum(k=0, n, (4*k+1)^(4*k+1)*x^k/(1+(4*k+1)^4*x +x*O(x^n))^(k+1)), n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(4*k+1)^(4*n+1))}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A222525, A220955, A221214, A217900, A007820, A242449.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 01 2013
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