|
%I #14 Oct 07 2020 08:39:31
%S 1,2,20,552,66896,33696416,68788184384,563088100346496,
%T 18447871370917745920,2417888544016592098109952,
%U 1267655436300759217689238066176,2658458526919399457630738994278213632
%N G.f. A(x) = Product_{n>=1} 1/(1 - 2^(n^2)*x^n).
%F G.f.: Sum_{n>=0} 2^(n^2) * x^n / Product_{k=1..n} (1 - 2^(k^2)*x^k).
%F G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} d*2^(n*d) ).
%F Logarithmic derivative yields A209803.
%F a(n) ~ 2^(n^2). - _Vaclav Kotesovec_, Oct 07 2020
%e G.f.: A(x) = 1 + 2*x + 20*x^2 + 552*x^3 + 66896*x^4 + 33696416*x^5 +...
%e such that the g.f. A(x) satisfies the identity:
%e 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)*...)
%e 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)) +...
%t nmax = 15; CoefficientList[Series[Product[1/(1 - 2^(k^2)*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 07 2020 *)
%o (PARI) a(n)=polcoeff(1/prod(k=1,n,1-2^(k^2)*x^k+x*O(x^n)),n)
%o (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)}
%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n,x^m/m*sumdiv(m, d, d*2^(m*d)))+x*O(x^n)),n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A000041, A209803 (log), A209495.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 17 2009
| 