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A209495
G.f. A(x) = Product_{n>=1} 1/(1 - 3^(n^2)*x^n).
3
1, 3, 90, 19953, 43113141, 847419543189, 150097181430365019, 239299779591639615799629, 3433684538204027455551495578190, 443426498544110770796791015548539667738, 515377522062291104801900194512917291134738021506
OFFSET
0,2
LINKS
FORMULA
G.f.: Sum_{n>=0} 3^(n^2) * x^n / Product_{k=1..n} (1 - 3^(k^2)*x^k).
G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} d*3^(n*d) ).
Logarithmic derivative yields A209804.
a(n) ~ 3^(n^2). - Vaclav Kotesovec, Oct 10 2020
EXAMPLE
G.f.: A(x) = 1 + 3*x + 90*x^2 + 19953*x^3 + 43113141*x^4 +...
such that the g.f. A(x) satisfies the identity:
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)*...)
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)) +...
MATHEMATICA
CoefficientList[Series[Product[1/(1 - 3^(n^2)*x^n), {n, 1, 1000}], {x, 0, 40}], x] (* G. C. Greubel, Mar 05 2018 *)
PROG
(PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-3^(k^2)*x^k+x*O(x^n)), n)}
(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)}
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, x^m/m*sumdiv(m, d, d*3^(m*d)))+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
Cf. A209804 (log), A157317.
Sequence in context: A013303 A376777 A166334 * A168408 A132556 A053996
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 09 2012
STATUS
approved