A157317 G.f. A(x) = Product_{n>=1} 1/(1 - 2^(n^2)*x^n).

1, 2, 20, 552, 66896, 33696416, 68788184384, 563088100346496, 18447871370917745920, 2417888544016592098109952, 1267655436300759217689238066176, 2658458526919399457630738994278213632

Offset

0,2

Links

Table of n, a(n) for n=0..11.

Formula

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

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} d*2^(n*d) ).

Logarithmic derivative yields A209803.

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Oct 07 2020

Example

G.f.: A(x) = 1 + 2*x + 20*x^2 + 552*x^3 + 66896*x^4 + 33696416*x^5 +...
such that the g.f. A(x) satisfies the identity:
A(x) = 1/((1-2*x)*(1-2^4*x^2)*(1-2^9*x^3)*(1-2^16*x^4)*(1-2^25*x^5)*...)
A(x) = 1 + 2*x/(1-2*x) + 2^4*x^2/((1-2*x)*(1-2^4*x^2)) + 2^9*x^3/((1-2*x)*(1-2^4*x^2)*(1-2^9*x^3)) + 2^16*x^4/((1-2*x)*(1-2^4*x^2)*(1-2^9*x^3)*(1-2^16*x^4)) +...

Mathematica

nmax = 15; CoefficientList[Series[Product[1/(1 - 2^(k^2)*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 07 2020 *)

Prog

(PARI) a(n)=polcoeff(1/prod(k=1, n, 1-2^(k^2)*x^k+x*O(x^n)), n)
(PARI) {a(n)=polcoeff(1+sum(m=1, n, 2^(m^2)*x^m/prod(k=1, m, 1-(2^k*x)^k+x*O(x^n))), n)}
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, x^m/m*sumdiv(m, d, d*2^(m*d)))+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A000041, A209803 (log), A209495.

Sequence in context: A009699 A101927 A341269 * A350794 A009399 A275779

Adjacent sequences: A157314 A157315 A157316 * A157318 A157319 A157320

Keyword

nonn

Author

Paul D. Hanna, Mar 17 2009

Status

approved