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%I #22 Jan 26 2017 01:29:12
%S 1,1,7,125,3641,148297,7792275,502572905,38466067169,3409770740129,
%T 343687137315215,38829855954523317,4861184771611069929,
%U 668044273723230765337,99988042875734734075243,16191529121372446646518737,2820684538705808192370559425
%N O.g.f.: Sum_{n>=0} n^n*(2*n+1)^(n-1) * exp(-n*(2*n+1)*x) * x^n / n!.
%C Compare the g.f. to the LambertW identity:
%C 1 = Sum_{n>=0} (2*n+1)^(n-1) * exp(-(2*n+1)*x) * x^n/n!.
%H G. C. Greubel, <a href="/A217910/b217910.txt">Table of n, a(n) for n = 0..315</a>
%F a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (2*k+1)^(n-1).
%F a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(2*k+1)^(k-1)*x^k / (1 + k*(2*k+1)*x)^(k+1).
%F a(n) = [x^n] 1 + x*(1+x)^(n-1) / Product_{k=1..n} (1 - 2*k*x).
%F a(n) = [x^n] 1 + x*(1-x)^(n-1) / Product_{k=1..n} (1 - (2*k+1)*x).
%F a(n) ~ 2^(3*n-9/4) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+1/4)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... = 2*A106533. - _Vaclav Kotesovec_, May 22 2014
%e O.g.f.: A(x) = 1 + x + 7*x^2 + 125*x^3 + 3641*x^4 + 148297*x^5 + 7792275*x^6 +...
%e where
%e A(x) = 1 + 1^1*3^0*x*exp(-1*3*x) + 2^2*5^1*exp(-2*5*x)*x^2/2! + 3^3*7^2*exp(-3*7*x)*x^3/3! + 4^4*9^3*exp(-4*9*x)*x^4/4! + 5^5*11^4*exp(-5*11*x)*x^5/5! +...
%e simplifies to a power series in x with integer coefficients.
%t Flatten[{1,Table[Sum[Binomial[n-1,j]*2^j*StirlingS2[n+j,n],{j,0,n-1}],{n,1,20}]}] (* _Vaclav Kotesovec_, May 22 2014 *)
%o (PARI) {a(n)=polcoeff(sum(k=0,n,k^k*(2*k+1)^(k-1)*x^k*exp(-k*(2*k+1)*x+x*O(x^n))/k!),n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, k^k*(2*k+1)^(k-1)*x^k/(1+k*(2*k+1)*x +x*O(x^n))^(k+1)), n)}
%o (PARI) {a(n)=1/n!*sum(k=0,n, (-1)^(n-k)*binomial(n,k)*k^n*(2*k+1)^(n-1))}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n)=polcoeff(1+x*(1+x)^(n-1)/prod(k=0, n, 1-2*k*x +x*O(x^n)), n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n)=polcoeff(1+x*(1-x)^n/prod(k=0, n, 1-(2*k+1)*x +x*O(x^n)), n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A217899, A217900, A217901, A217902, A217903, A217905, A217911.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 14 2012
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