A316142 -id:A316142 - cp's OEIS frontend

A316143 Expansion of e.g.f. Product_{k>=1} 1 / (1 - (exp(x)-1)^k)^2.

1, 2, 12, 92, 912, 10772, 148512, 2328692, 40842912, 791302772, 16767551712, 385382491892, 9542377300512, 253105962752372, 7156766466076512, 214814484529608692, 6819311473596695712, 228212485803422931572, 8028037725386962194912, 296094910181041530831092

Offset: 0

Vaclav Kotesovec, Jun 25 2018

Self-convolution of A167137.

Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 7 we obtain the sequence [1, 2, 5, 1, 2, 6, 0, 2, 5, 1, 2, 6, 0, 2, 5, 1, 2, 6, 0, ...], with a preperiod of length 1 and an apparent period thereafter of 6 = phi(7). - Peter Bala, Mar 03 2023

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A000712, A167137, A316142, A316144.

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

Sum_{k=0..n} binomial(n,k) * A167137(k) * A167137(n-k).

a(n) ~ n! * exp(Pi * sqrt(2*n/(3*log(2))) - Pi^2 * (1 - 1/log(2)) / 12) / (2^(7/4) * 3^(3/4) * n^(5/4) * (log(2))^(n - 1/4)).

A316144 Expansion of e.g.f. Product_{k>=1} ((1 + (exp(x)-1)^k) / (1 - (exp(x)-1)^k))^2.

Original entry on oeis.org

1, 4, 28, 268, 3148, 43564, 692428, 12390508, 245896588, 5351817004, 126614238028, 3232332423148, 88500275727628, 2585371577628844, 80227707005300428, 2634361286274638188, 91223969834203056268, 3321457538305952791084, 126817592900018186967628

Offset: 0

Vaclav Kotesovec, Jun 25 2018

Self-convolution of A306045.
Convolution of A316142 and A316143.
Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 7 we obtain the sequence [1, 4, 0, 2, 5, 3, 2, 4, 0, 2, 5, 3, 2, 4, 0, 2, 5, 3, 2, ...], with a preperiod of length 1 and an apparent period thereafter of 6 = phi(7). - Peter Bala, Mar 03 2023

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A001934, A306045, A316142, A316143.

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

Sum_{k=0..n} binomial(n,k) * A306045(k) * A306045(n-k).

a(n) ~ n! * exp(Pi * sqrt(n/log(2)) - Pi^2 * (1 - 1/log(2)) / 8) / (2^(7/2) * n^(5/4) * log(2)^(n - 1/4)).

cp's OEIS Frontend

A316143 Expansion of e.g.f. Product_{k>=1} 1 / (1 - (exp(x)-1)^k)^2.

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