A235385 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.

1, 1, 1, 4, 13, 75, 415, 3160, 24545, 233509, 2323165, 26599780, 321545365, 4312503655, 61219938915, 942271981240, 15340303899265, 266671144108265, 4892612440317145, 94840103781865060, 1934826541931748925, 41387703314570495875, 928953515444722956775, 21738929496091877729400

Offset: 0

Author

Paul D. Hanna, Jan 08 2014

Keywords

• nonn

Comments

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

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 75*x^5/5! +...
where
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 +...
Explicitly,
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 +...
Equivalently,
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! +...
		

Crossrefs

Cf. A235685, A087761.

Programs

PARI

{H(n)=sum(k=1,n,1/k)}
{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)}
for(n=0,25,print1(a(n),", "))