A306083 -id:A306083 - 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)).

A306082 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^(k^2)).

Original entry on oeis.org

1, 1, 3, 13, 99, 901, 8763, 92653, 1125939, 16333141, 274594923, 5041348093, 97841114979, 2007694705381, 44043941312283, 1036207737976333, 25969433606691219, 688418684249653621, 19275116061819888843, 571069469474068377373, 17898523203378840958659

Offset: 0

Vaclav Kotesovec, Jun 20 2018

nonn

Conjecture: for positive integer k, reducing the sequence modulo k produces an eventually periodic sequence with period dividing phi(k) = A000010(k). For example, modulo 7 we obtain the sequence [1, 1, 3, 6, 1, 5, 6, 1, 3, 6, 1, 5, 6, 1, 3, 6, 1, 5, 6, ...], with an apparent period of 6 beginning at a(1). - Peter Bala, Feb 22 2025

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

Cf. A001156, A294529, A306083, A306147.

a:=series(mul(1/(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/(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) * A001156(k) * k!.

a(n) ~ n! * exp(3 * 2^(-5/3) * Zeta(3/2)^(2/3) * (Pi*n/log(2))^(1/3)) * Zeta(3/2)^(2/3) / (2^(13/6) * sqrt(3) * Pi^(7/6) * n^(7/6) * (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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