A235685 - OEIS

%I #6 Sep 06 2024 15:08:22

%S 1,1,12,370,21490,2012346,276603096,52410015372,13086020092860,

%T 4162089324641820,1642246641208135248,786999935144858519448,

%U 450157417640763926225496,302899153448396612425831800,236824761815538353605549389600,212892105513043495761147435785040

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

%C Compare to: exp( Sum_{n>=1} x^(2*n)/(2*n) ) = 1/sqrt(1-x^2).

%e E.g.f.: A(x) = 1 + x^2/2! + 12*x^4/4! + 370*x^6/6! + 21490*x^8/8! +...

%e where

%e 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 Explicitly,

%e 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 (PARI) {H(n)=sum(k=1, n, 1/k)}

%o {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 for(n=0, 25, print1(a(2*n), ", "))

%Y  Cf. A235385.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 13 2014