A306082 -id:A306082 - cp's OEIS frontend

A306147 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x)-1)^(k^2)) / (1 - (exp(x)-1)^(k^2)).

1, 2, 6, 26, 198, 2042, 22566, 259226, 3249798, 47156282, 799108326, 15116875226, 305203728198, 6488119430522, 146602455461286, 3557921474016026, 92563621667899398, 2554423824661976762, 74142584637465337446, 2258422219660738881626, 72255096004023644467398

Offset: 0

Vaclav Kotesovec, Jun 23 2018

nonn

Convolution of A306082 and A306083.

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Cf. A001156, A033461, A103265, A306082, A306083.

nmax = 20; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^(k^2)) / (1 - (Exp[x] - 1)^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = Sum_{k=0..n} Stirling2(n,k) * A103265(k) * k!.

a(n) ~ n! * ((2*sqrt(2) - 1) * Zeta(3/2))^(2/3) * exp(3 * (Pi/log(2))^(1/3) * ((2*sqrt(2) - 1) * Zeta(3/2))^(2/3) * n^(1/3) / 4) / (8 * sqrt(3) * Pi^(7/6) * n^(7/6) * (log(2))^(n - 1/6)).

A306083 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^(k^2)).

Original entry on oeis.org

1, 1, 1, 1, 25, 361, 3361, 25201, 166825, 1383481, 25879921, 651816001, 14450460025, 280347467401, 5253918022081, 107822784560401, 2578135250199625, 69030779356572121, 1953531819704493841, 56903093167217522401, 1689294590583626265625

Offset: 0

Vaclav Kotesovec, Jun 20 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Cf. A033461, A306082, A306147.

a:=series(mul(1+(exp(x)-1)^(k^2),k=1..100),x=0,21): seq(n!*coeff(a, x, n),n=0..20); # Paolo P. Lava, Mar 26 2019

nmax = 20; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = Sum_{k=0..n} Stirling2(n,k) * A033461(k) * k!.

a(n) ~ n! * exp(3 * (Pi/log(2))^(1/3) * ((sqrt(2) - 1) * Zeta(3/2))^(2/3) * n^(1/3) / 4) * ((sqrt(2) - 1) * Zeta(3/2) / Pi)^(1/3) / (2 * sqrt(6) * n^(5/6) * log(2)^(n + 1/6)).

cp's OEIS Frontend

A306147 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x)-1)^(k^2)) / (1 - (exp(x)-1)^(k^2)).

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