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 A283119 Expansion of exp( Sum_{n>=1} sigma(6*n)*x^n/n ) in powers of x. 6
 1, 12, 86, 469, 2141, 8594, 31247, 104945, 330094, 982284, 2786861, 7584060, 19893185, 50494558, 124437410, 298555264, 699017259, 1600364304, 3589048673, 7896510620, 17067607791, 36283650153, 75947406513, 156672628539, 318804641925, 640390347979 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS sigma(6*n) = A000203(6*n), the sum of divisors of 6*n (A224613). LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1000 FORMULA G.f.: Product_{n>=1} (1 - x^(2*n))^4 * (1 - x^(3*n))^3/((1 - x^n)^12 * (1 - x^(6*n))). a(n) = (1/n)*Sum_{k=1..n} sigma(6*k)*a(n-k). - Seiichi Manyama, Mar 05 2017 a(n) ~ 55^(7/4) * exp(sqrt(55*n)*Pi/3) / (41472*sqrt(3)*n^(9/4)). - Vaclav Kotesovec, Mar 20 2017 EXAMPLE G.f.: A(x) = 1 + 12*x + 86*x^2 + 469*x^3 + 2141*x^4 + 8594*x^5 + ... log(A(x)) = 12*x + 28*x^2/2 + 39*x^3/3 + 60*x^4/4 + 72*x^5/5 + 91*x^6/6 + 96*x^7/7 + 124*x^8/8 + ... + sigma(6*n)*x^n/n + ... MATHEMATICA Table[SeriesCoefficient[Product[(1 - x^(2 i))^4*(1 - x^(3 i))^3/((1 - x^i)^12*(1 - x^(6 i))), {i, n}], {x, 0, n}], {n, 0, 25}] (* Michael De Vlieger, Mar 01 2017 *) CROSSREFS Cf. A224613 (sigma(6*n)), A283164 (exp( Sum_{n>=1} -sigma(6*n)*x^n/n )). Cf. A182818 (k=2), A182819 (k=3), A182820 (k=4), A182821 (k=5), this sequence (k=6), A283077 (k=7), A283120 (k=8), A283121 (k=9). Sequence in context: A225785 A098206 A104911 * A091119 A243248 A046023 Adjacent sequences: A283116 A283117 A283118 * A283120 A283121 A283122 KEYWORD nonn AUTHOR Seiichi Manyama, Mar 01 2017 STATUS approved

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Last modified August 14 18:32 EDT 2024. Contains 375166 sequences. (Running on oeis4.)