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 A157317 G.f. A(x) = Product_{n>=1} 1/(1 - 2^(n^2)*x^n). 2
 1, 2, 20, 552, 66896, 33696416, 68788184384, 563088100346496, 18447871370917745920, 2417888544016592098109952, 1267655436300759217689238066176, 2658458526919399457630738994278213632 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS 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 * A009399 A275779 A292415 Adjacent sequences:  A157314 A157315 A157316 * A157318 A157319 A157320 KEYWORD nonn AUTHOR Paul D. Hanna, Mar 17 2009 STATUS approved

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Last modified June 21 01:08 EDT 2021. Contains 345337 sequences. (Running on oeis4.)