A209495 - OEIS

%I #15 Oct 10 2020 08:14:16

%S 1,3,90,19953,43113141,847419543189,150097181430365019,

%T 239299779591639615799629,3433684538204027455551495578190,

%U 443426498544110770796791015548539667738,515377522062291104801900194512917291134738021506

%N G.f. A(x) = Product_{n>=1} 1/(1 - 3^(n^2)*x^n).

%H G. C. Greubel, <a href="/A209495/b209495.txt">Table of n, a(n) for n = 0..45</a>

%F G.f.: Sum_{n>=0} 3^(n^2) * x^n / Product_{k=1..n} (1 - 3^(k^2)*x^k).

%F G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} d*3^(n*d) ).

%F Logarithmic derivative yields A209804.

%F a(n) ~ 3^(n^2). - _Vaclav Kotesovec_, Oct 10 2020

%e G.f.: A(x) = 1 + 3*x + 90*x^2 + 19953*x^3 + 43113141*x^4 +...

%e such that the g.f. A(x) satisfies the identity:

%e A(x) = 1/((1-3*x)*(1-3^4*x^2)*(1-3^9*x^3)*(1-3^16*x^4)*(1-3^25*x^5)*...)

%e A(x) = 1 + 3*x/(1-3*x) + 3^4*x^2/((1-3*x)*(1-3^4*x^2)) + 3^9*x^3/((1-3*x)*(1-3^4*x^2)*(1-3^9*x^3)) + 3^16*x^4/((1-3*x)*(1-3^4*x^2)*(1-3^9*x^3)*(1-3^16*x^4)) +...

%t CoefficientList[Series[Product[1/(1 - 3^(n^2)*x^n), {n, 1, 1000}], {x, 0, 40}], x] (* _G. C. Greubel_, Mar 05 2018 *)

%o (PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-3^(k^2)*x^k+x*O(x^n)), n)}

%o (PARI) {a(n)=polcoeff(1+sum(m=1, n, 3^(m^2)*x^m/prod(k=1, m, 1-(3^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*3^(m*d)))+x*O(x^n)),n)}

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

%Y Cf. A209804 (log), A157317.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 09 2012