A156171 - OEIS

%I #9 Oct 17 2020 08:25:21

%S 1,1,3,11,53,357,3521,51665,1122135,35638903,1639453459,108526044099,

%T 10298220348807,1396920580458279,270394562069007327,

%U 74574294532698008703,29276455806256470979269,16344863466384180848085765,12969208162308705691408055345,14616452655308018025267503353697

%N G.f.: A(x) = exp( Sum_{n>=1} x^n/(1 - 2^n*x)^n / n ), a power series in x with integer coefficients.

%H Paul D. Hanna, <a href="/A156171/b156171.txt">Table of n, a(n) for n = 0..150</a>

%F a(n) ~ c * 2^(n^2/4 + n + 1/2) / (sqrt(Pi) * n^(3/2)), where c = EllipticTheta[3, 0, 1/2] = JacobiTheta3(0,1/2) = 2.1289368272118771586694585... if n is even and c = EllipticTheta[2, 0, 1/2] = JacobiTheta2(0,1/2) = 2.1289312505130275585916134... if n is odd. - _Vaclav Kotesovec_, Oct 17 2020

%e G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 53*x^4 + 357*x^5 + 3521*x^6 + 51665*x^7 + 1122135*x^8 + 35638903*x^9 + 1639453459*x^10 + 108526044099*x^11 +...

%e such that:

%e log(A(x)) = Sum_{n>=1} x^n/n * (1 + 2^n*x + 4^n*x^2 +...+ 2^(n*k)*x^k +...)^n

%e or

%e log(A(x)) = x*(1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + 32*x^5 +...) +

%e x^2/2*(1 + 8*x + 48*x^2 + 256*x^3 + 1280*x^4 + 6144*x^5 +...) +

%e x^3/3*(1 + 24*x + 384*x^2 + 5120*x^3 + 61440*x^4 + 688128*x^5 +...) +

%e x^4/4*(1 + 64*x + 2560*x^2 + 81920*x^3 + 2293760*x^4 + 58720256*x^5 +...) +

%e x^5/5*(1 + 160*x + 15360*x^2 + 1146880*x^3 + 73400320*x^4 + 4227858432*x^5 +...) +

%e x^6/6*(1 + 384*x + 86016*x^2 + 14680064*x^3 + 2113929216*x^4 + 270582939648*x^5 +...) +...

%e Explicitly,

%e log(A(x)) = x + 5*x^2/2 + 25*x^3/3 + 161*x^4/4 + 1441*x^5/5 + 18305*x^6/6 + 330625*x^7/7 + 8488961*x^8/8 + 309465601*x^9/9 + 16011372545*x^10/10 + 1174870185985*x^11/11 + 122233833963521*x^12/12 +...

%t nmax = 20; CoefficientList[Series[Exp[Sum[x^k/(1 - 2^k*x)^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 17 2020 *)

%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n,x^m/(1-2^m*x+x*O(x^n))^m/m)),n)}

%Y Cf. A156170, A155200, A156100.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 05 2009