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%I #9 Nov 01 2014 04:42:28
%S 1,5,85,3389,238021,25791485,3982831525,830287473629,224589628828741,
%T 76476839514843965,32008234421462900965,16149704161940128467869,
%U 9666556369631455442262661,6772084092710344017948232445,5489289305870251277606850778405
%N G.f.: Sum_{n>=0} (2*n+1)! * x^n / Product_{k=1..2*n+1} (1 + k*x).
%C Compare g.f. to: 1/(1-x) = Sum_{n>=0} n!*x^n/Product_{k=1..n} (1+k*x).
%H Vaclav Kotesovec, <a href="/A208886/b208886.txt">Table of n, a(n) for n = 0..210</a>
%F a(n) ~ sqrt(Pi) * 2^(2*n+2) * n^(2*n+3/2) / exp(2*n+1/2). - _Vaclav Kotesovec_, Nov 01 2014
%e G.f.: A(x) = 1 + 5*x + 85*x^2 + 3389*x^3 + 238021*x^4 + 25791485*x^5 +...
%e such that
%e A(x) = 1/(1+x) + 3!*x/((1+x)*(1+2*x)*(1+3*x)) + 5!*x^2/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)*(1+5*x)) + 7!*x^3/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)*(1+5*x)*(1+6*x)*(1+7*x)) + 9!*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)*(1+5*x)*(1+6*x)*(1+7*x)*(1+8*x)*(1+9*x)) +...
%o (PARI) {a(n)=polcoeff(sum(m=0,n,(2*m+1)!*x^m/prod(k=1,2*m+1,1+k*x+x*O(x^n))),n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A208885, A189919.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 04 2012
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