A235776 - OEIS

%I #10 Oct 28 2024 17:20:08

%S 1,2,42,2000,170660,22741992,4344779208,1123066676160,376718037181200,

%T 158895919895100960,82222168141278271392,51172838316787466103552,

%U 37687233953299944682503744,32399590493755848692815785600,32140659218911596667452247171200

%N E.g.f.: exp( Sum_{n>=1} x^(2*n) * Sum_{k=1..n} 1/k^2 ).

%H Vaclav Kotesovec, <a href="/A235776/b235776.txt">Table of n, a(n) for n = 0..220</a>

%e E.g.f.: A(x) = 1 + 2*x^2/2! + 42*x^4/4! + 2000*x^6/6! + 170660*x^8/8! +...

%e such that

%e log(A(x)) = x^2 + (1+1/4)*x^4 + (1+1/4+1/9)*x^6 + (1+1/4+1/9+1/16)*x^8 + (1+1/4+1/9+1/16+1/25)*x^10 + (1+1/4+1/9+1/16+1/25+1/36)*x^12 +...

%e Explicitly,

%e log(A(x)) = x^2 + 5/4*x^4 + 49/36*x^6 + 205/144*x^8 + 5269/3600*x^10 + 5369/3600*x^12 + 266681/176400*x^14 +...+ [Sum_{k=1..n} 1/k^2]*x^(2*n) +...

%t nmax = 20; CoefficientList[Series[Exp[PolyLog[2,x]/(1-x)], {x, 0, nmax}], x] * (2*Range[0, nmax])! (* _Vaclav Kotesovec_, Oct 28 2024 *)

%o (PARI) {a(n)=local(A=1); A=exp(sum(m=1, n\2+1, sum(k=1, m, 1/k^2)*x^(2*m))+x*O(x^n)); n!*polcoeff(A, n)}

%o for(n=0, 20, print1(a(2*n), ", "))

%Y Cf. A087761, A235385.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 15 2014