A207654 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(2*k-1) - 1)/(1 - x^(2*k-1)).

1, 1, 4, 22, 173, 1816, 23659, 367573, 6622465, 135637477, 3111148862, 78984029782, 2198423489832, 66562555228478, 2177861372888738, 76571625673934064, 2878937040339348981, 115260759545001030638, 4895471242828376133806, 219853190410155476470763

Offset: 0

Author

Paul D. Hanna, Feb 19 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 173*x^4 + 1816*x^5 + 23659*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)-1)/(1-x) + ((1+x)-1)*((1+x)^3-1)/((1-x)*(1-x^3)) + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)/((1-x)*(1-x^3)*(1-x^5)) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200
• Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Crossrefs

Cf. A207651, A207652, A207653.

Programs

Mathematica

With[{nn=20},CoefficientList[Series[Sum[Product[((1+x)^(2k-1)-1)/(1- x^(2k-1)),{k,n}],{n,0,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Sep 06 2015 *)

PARI

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,((1+x)^(2*k-1)-1)/(1-x^(2*k-1) +x*O(x^n)) )),n)}
for(n=0,25,print1(a(n),", "))

Formula

From Vaclav Kotesovec, Oct 31 2014: (Start)

a(n) ~ sqrt(6) * 24^n * n! / (exp(Pi^2/48) * sqrt(n) * Pi^(2*n+3/2)).

a(n) ~ 2^n * 12^(n+1/2) * n^n / (exp(n + Pi^2/48) * Pi^(2*n+1)).

(End)