A181315 G.f. A(x) satisfies A(x) = Product_{n>=1} (1 + x^n*A(x)^n).
1, 1, 2, 6, 19, 64, 227, 832, 3125, 11970, 46579, 183614, 731688, 2942673, 11928707, 48688888, 199932987, 825379993, 3423614756, 14261439594, 59635806865, 250241613688, 1053380320889, 4446989542144, 18823433444211, 79871578901283
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 227*x^6 +... The g.f. A = A(x) satisfies log(A) = x*A/(1-x^2*A^2) + (x^2/2)*A^2/(1-x^4*A^4) + (x^3/3)*A^3/(1-x^6*A^6) +...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Programs
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Maple
nmax:=25: kmax:=nmax: for n from 1 to nmax+1 do A(x):=add(a(k)*x^k, k=0..kmax-1): A(x) := product((1 + x^k*A(x)^k),k=1..kmax+1): a(n-1):=coeff(A(x),x,n-1): od: seq(a(n),n=0..nmax); # Johannes W. Meijer, Jul 04 2011
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Mathematica
InverseSeries[x QPochhammer[x, x^2] + O[x]^30][[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)
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PARI
{a(n)=polcoeff(1/x*serreverse(x/prod(k=1,n+1,1+x^k+x*O(x^n))),n)} -
PARI
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (x*A+x*O(x^n))^m/(1-(x*A)^(2*m))/m))); polcoeff(A, n)}
Formula
G.f.: A(x) = Sum_{n>=0} A000009(n)*x^n*A(x)^n, where A000009(n) is the number of partitions of n into distinct parts.
G.f.: A(x) = (1/x)*Series_Reversion[x^(1/24)*eta(x)/eta(x^2)] (cf. A081362).
G.f. satisfies A(x) = exp( Sum_{n>=1} (x^n/n)*A(x)^n/(1 - (x*A(x))^(2*n)) ).
a(n) ~ c * d^n / n^(3/2), where d = A270914 = 4.50247674761735448773859393270078440676312875609162163346454... and c = A366018 = 0.482420439587319764659364391266849418507665645926542970519109122... - Vaclav Kotesovec, Aug 21 2018