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%I #7 Jan 13 2014 22:02:00
%S 1,1,1,4,13,75,415,3160,24545,233509,2323165,26599780,321545365,
%T 4312503655,61219938915,942271981240,15340303899265,266671144108265,
%U 4892612440317145,94840103781865060,1934826541931748925,41387703314570495875,928953515444722956775,21738929496091877729400
%N E.g.f.: exp( Sum_{n>=1} H(n) * x^(2*n-1)/(2*n-1) ) where H(n) is the n-th harmonic number.
%C Compare to: exp( Sum_{n>=1} x^(2*n-1)/(2*n-1) ) = sqrt(1-x^2)/(1-x).
%e E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 75*x^5/5! +...
%e where
%e log(A(x)) = x + (1+1/2)*x^3/3 + (1+1/2+1/3)*x^5/5 + (1+1/2+1/3+1/4)*x^7/7 + (1+1/2+1/3+1/4+1/5)*x^9/9 + (1+1/2+1/3+1/4+1/5+1/6)*x^11/11 +...
%e Explicitly,
%e log(A(x)) = x + 1/2*x^3 + 11/30*x^5 + 25/84*x^7 + 137/540*x^9 + 49/220*x^11 + 363/1820*x^13 + 761/4200*x^15 +...
%e Equivalently,
%e log(A(x)) = x + 3*x^3/3! + 44*x^5/5! + 1500*x^7/7! + 92064*x^9/9! + 8890560*x^11/11! + 1241982720*x^13/13! + 236938141440*x^15/15! +...
%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-1)/(2*k-1))+x*O(x^n));n!*polcoeff(A,n)}
%o for(n=0,25,print1(a(n),", "))
%Y Cf. A235685, A087761.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jan 08 2014
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