cp's OEIS Frontend

A235685 E.g.f.: exp( Sum_{n>=1} H(n) * x^(2*n)/(2*n) ) where H(n) is the n-th harmonic number.

%I A235685 #6 Sep 06 2024 15:08:22
%S A235685 1,1,12,370,21490,2012346,276603096,52410015372,13086020092860,
%T A235685 4162089324641820,1642246641208135248,786999935144858519448,
%U A235685 450157417640763926225496,302899153448396612425831800,236824761815538353605549389600,212892105513043495761147435785040
%N A235685 E.g.f.: exp( Sum_{n>=1} H(n) * x^(2*n)/(2*n) ) where H(n) is the n-th harmonic number.
%C A235685 Compare to: exp( Sum_{n>=1} x^(2*n)/(2*n) ) = 1/sqrt(1-x^2).
%e A235685 E.g.f.: A(x) = 1 + x^2/2! + 12*x^4/4! + 370*x^6/6! + 21490*x^8/8! +...
%e A235685 where
%e A235685 log(A(x)) = x^2/2 + (1+1/2)*x^4/4 + (1+1/2+1/3)*x^6/6 + (1+1/2+1/3+1/4)*x^8/8 + (1+1/2+1/3+1/4+1/5)*x^10/10 + (1+1/2+1/3+1/4+1/5+1/6)*x^12/12 +...
%e A235685 Explicitly,
%e A235685 log(A(x)) = x^2/2! + 9*x^4/4! + 220*x^6/6! + 10500*x^8/8! + 828576*x^10/10! + 97796160*x^12/12! + 16145775360*x^14/14! + 3554072121600*x^16/16! +...
%o A235685 (PARI) {H(n)=sum(k=1, n, 1/k)}
%o A235685 {a(n)=local(A=1); A=exp(sum(k=1, n\2+1, H(k)*x^(2*k)/(2*k))+x*O(x^n)); n!*polcoeff(A, n)}
%o A235685 for(n=0, 25, print1(a(2*n), ", "))
%Y A235685  Cf. A235385.
%K A235685 nonn
%O A235685 0,3
%A A235685 _Paul D. Hanna_, Jan 13 2014