A283119 Expansion of exp( Sum_{n>=1} sigma(6*n)*x^n/n ) in powers of x.

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

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