cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

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%I A207654 #17 Feb 07 2020 13:38:40
%S A207654 1,1,4,22,173,1816,23659,367573,6622465,135637477,3111148862,
%T A207654 78984029782,2198423489832,66562555228478,2177861372888738,
%U A207654 76571625673934064,2878937040339348981,115260759545001030638,4895471242828376133806,219853190410155476470763
%N A207654 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(2*k-1) - 1)/(1 - x^(2*k-1)).
%H A207654 Vaclav Kotesovec, <a href="/A207654/b207654.txt">Table of n, a(n) for n = 0..200</a>
%H A207654 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 A207654 From _Vaclav Kotesovec_, Oct 31 2014: (Start)
%F A207654 a(n) ~ sqrt(6) * 24^n * n! / (exp(Pi^2/48) * sqrt(n) * Pi^(2*n+3/2)).
%F A207654 a(n) ~ 2^n * 12^(n+1/2) * n^n / (exp(n + Pi^2/48) * Pi^(2*n+1)).
%F A207654 (End)
%e A207654 G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 173*x^4 + 1816*x^5 + 23659*x^6 +...
%e A207654 such that, by definition,
%e A207654 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 A207654 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 A207654 (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 A207654 for(n=0,25,print1(a(n),", "))
%Y A207654 Cf. A207651, A207652, A207653.
%K A207654 nonn
%O A207654 0,3
%A A207654 _Paul D. Hanna_, Feb 19 2012