A341506 -id:A341506 - cp's OEIS frontend

A341505 E.g.f.: Product_{i>=1, j>=1} 1 / (1 - x^(ij) / (ij)!).

1, 1, 4, 14, 77, 427, 3076, 23088, 205316, 1936275, 20611750, 233576818, 2909340750, 38527889389, 551372037898, 8364582709282, 135560933977809, 2320127265064403, 42072789623722518, 802547153889643250, 16118882845967168807, 339268639052195731063

Offset: 0

Ilya Gutkovskiy, Feb 13 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..447

Cf. A000005, A005651, A006171, A174661, A318693, A340903, A341506, A341876.

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k/k!)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[d DivisorSigma[0, d]/(d!)^(k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 21}]

E.g.f.: Product_{k>=1} 1 / (1 - x^k / k!)^sigma_0(k).

a(n) ~ c * n!, where c = Product_{k>=2} 1/(1 - 1/k!)^sigma_0(k) = 6.6953800201104498115311861820134227776761182601282551253439990653959... - Vaclav Kotesovec, Feb 20 2021

A341876 E.g.f.: Product_{i>=1, j>=1} (1 + x^(ij)/(ij)!) / (1 - x^(ij)/(ij)!).

Original entry on oeis.org

1, 2, 8, 40, 230, 1584, 12096, 103828, 975284, 10045182, 111724064, 1342990356, 17288290776, 238095398064, 3488772309480, 54304690352816, 894465560384026, 15564259644205288, 285282543243628356, 5498843253154821196, 111203939051325462504, 2355689449259544720344

Offset: 0

Vaclav Kotesovec, Feb 22 2021

nonn

Exponential convolution of A341505 and A341506.

Vaclav Kotesovec, Table of n, a(n) for n = 0..447

Cf. A341505, A341506.

nmax = 25; CoefficientList[Series[Product[(1 + x^(i*j)/(i*j)!)/(1 - x^(i*j)/(i*j)!), {i, 1, nmax}, {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 25; CoefficientList[Series[Product[((1 + x^k/k!)/(1 - x^k/k!))^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]*Range[0, nmax]!

a(n) = Sum_{k=0..n} binomial(n,k) * A341505(k) * A341506(n-k).

a(n) ~ c * n!, where c = 2*Product_{k>=2} ((k!+1)/(k!-1))^sigma_0(k) = 47.4139841600096613008093034069984807541890052309118213077603602425211186...

cp's OEIS Frontend

A341505 E.g.f.: Product_{i>=1, j>=1} 1 / (1 - x^(ij) / (ij)!).

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A341876 E.g.f.: Product_{i>=1, j>=1} (1 + x^(ij)/(ij)!) / (1 - x^(ij)/(ij)!).

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cp's OEIS Frontend

A341505 E.g.f.: Product_{i>=1, j>=1} 1 / (1 - x^(i*j) / (i*j)!).

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A341876 E.g.f.: Product_{i>=1, j>=1} (1 + x^(i*j)/(i*j)!) / (1 - x^(i*j)/(i*j)!).

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A341505 E.g.f.: Product_{i>=1, j>=1} 1 / (1 - x^(ij) / (ij)!).

A341876 E.g.f.: Product_{i>=1, j>=1} (1 + x^(ij)/(ij)!) / (1 - x^(ij)/(ij)!).