A207654 - OEIS

%I #17 Feb 07 2020 13:38:40

%S 1,1,4,22,173,1816,23659,367573,6622465,135637477,3111148862,

%T 78984029782,2198423489832,66562555228478,2177861372888738,

%U 76571625673934064,2878937040339348981,115260759545001030638,4895471242828376133806,219853190410155476470763

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

%H Vaclav Kotesovec, <a href="/A207654/b207654.txt">Table of n, a(n) for n = 0..200</a>

%H Hsien-Kuei Hwang, Emma Yu Jin, <a href="https://arxiv.org/abs/1911.06690">Asymptotics and statistics on Fishburn matrices and their generalizations</a>, arXiv:1911.06690 [math.CO], 2019.

%F From _Vaclav Kotesovec_, Oct 31 2014: (Start)

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

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

%F (End)

%e G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 173*x^4 + 1816*x^5 + 23659*x^6 +...

%e such that, by definition,

%e 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)) +...

%t 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 *)

%o (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)}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A207651, A207652, A207653.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 19 2012