A156171 G.f.: A(x) = exp( Sum_{n>=1} x^n/(1 - 2^n*x)^n / n ), a power series in x with integer coefficients.

1, 1, 3, 11, 53, 357, 3521, 51665, 1122135, 35638903, 1639453459, 108526044099, 10298220348807, 1396920580458279, 270394562069007327, 74574294532698008703, 29276455806256470979269, 16344863466384180848085765, 12969208162308705691408055345, 14616452655308018025267503353697

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..150

Formula

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

Example

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 +...
such that:
log(A(x)) = Sum_{n>=1} x^n/n * (1 + 2^n*x + 4^n*x^2 +...+ 2^(n*k)*x^k +...)^n
or
log(A(x)) = x*(1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + 32*x^5 +...) +
x^2/2*(1 + 8*x + 48*x^2 + 256*x^3 + 1280*x^4 + 6144*x^5 +...) +
x^3/3*(1 + 24*x + 384*x^2 + 5120*x^3 + 61440*x^4 + 688128*x^5 +...) +
x^4/4*(1 + 64*x + 2560*x^2 + 81920*x^3 + 2293760*x^4 + 58720256*x^5 +...) +
x^5/5*(1 + 160*x + 15360*x^2 + 1146880*x^3 + 73400320*x^4 + 4227858432*x^5 +...) +
x^6/6*(1 + 384*x + 86016*x^2 + 14680064*x^3 + 2113929216*x^4 + 270582939648*x^5 +...) +...
Explicitly,
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 +...

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A156170, A155200, A156100.

Sequence in context: A321732 A224345 A081367 * A129093 A259106 A057325

Adjacent sequences: A156168 A156169 A156170 * A156172 A156173 A156174

Keyword

nonn

Author

Paul D. Hanna, Feb 05 2009

Status

approved