A194353 G.f.: exp( Sum_{n>=1} (x^n/n)/sqrt(1 - 4*x^n) ).
1, 1, 3, 9, 32, 114, 431, 1633, 6306, 24466, 95604, 374920, 1475731, 5822941, 23027817, 91225983, 361945480, 1437855016, 5718242653, 22762417519, 90684186378, 361540944078, 1442313708600, 5757140994990, 22991764612516, 91861397186492
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 32*x^4 + 114*x^5 + 431*x^6 +... where log(A(x)) = x/sqrt(1-4*x) + (x^2/2)/sqrt(1-4*x^2) + (x^3/3)/sqrt(1-4*x^3) +... Other expansions: log(A(x)) = x*(1 + 2*x + 6*x^2 + 20*x^3 + 70*x^4 + 252*x^5 +...) + x^2/2*(1 + 2*x^2 + 6*x^4 + 20*x^6 + 70*x^8 + 252*x^10 +...) + x^3/3*(1 + 2*x^3 + 6*x^6 + 20*x^9 + 70*x^12 + 252*x^15 +...) +... log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 85*x^4/4 + 351*x^5/5 + 1535*x^6/6 + 6469*x^7/7 +... 1/A(x) = (1-x)*(1-x^2)^2*(1-x^3)^6*(1-x^4)^20*(1-x^5)^70*(1-x^6)^252*...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Programs
-
Mathematica
nmax = 25; CoefficientList[Series[Exp[Sum[x^j/j/Sqrt[1 - 4*x^j], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 10 2021 *) nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^Binomial[2*k - 2, k - 1], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 10 2021 *) -
PARI
{a(n)=polcoeff(exp(sum(m=1,n+1,(x^m/m)/sqrt(1-4*x^m+x*O(x^n)))),n)}
Formula
G.f.: Product_{n>=1} 1/(1-x^n)^binomial(2*n-2,n-1).
a(n) ~ 2^(2*n-1) * exp(3*n^(1/3)/4 + c) / (sqrt(3*Pi) * n^(5/6)), where c = Sum_{k>=2} 1/(k * 4^k * sqrt(1 - 4^(1-k))) = 0.0426953980609366317948752143053832860841846208508627487697950... - Vaclav Kotesovec, May 10 2021