A174465 -id:A174465 - cp's OEIS frontend

A321191 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^tau_n(k), where tau_n(k) = number of ordered n-factorizations of k.

1, 1, 3, 7, 29, 71, 336, 932, 4593, 13690, 69708, 222718, 1163734, 3902016, 20825927, 73229397, 397806717, 1452193925, 8016518379, 30328368519, 169781766056, 662143701506, 3755514158949, 15071604241851, 86496856963200, 356063096545571, 2066351471542036

Offset: 0

Ilya Gutkovskiy, Oct 29 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A000041, A006171, A077592, A174465, A280487, A321192.

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[#, k-1] & /@ Divisors[n]); nmax = 30; Table[SeriesCoefficient[Product[1/(1 - x^k)^tau[k, n], {k, 1, n}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)

a(n) = [x^n] Product_{k_1>=1, k_2>=1, ..., k_n>=1} 1/(1 - x^(k_1*k_2*...*k_n)).

A318966 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} 1/(1 - x^(ijk))^(1/(ijk)).

Original entry on oeis.org

1, 1, 5, 21, 165, 1077, 11457, 103905, 1345257, 15834825, 237535389, 3372509709, 59235634125, 979573962429, 19224990899865, 366788042231193, 8019002662543953, 171360055378885905, 4132946756763614133, 97947895990285022085, 2576516749059849502581, 67124117357620005459141

Offset: 0

Ilya Gutkovskiy, Sep 06 2018

nonn

Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000005, A007425, A007426, A028342, A174465, A318413, A318695, A318967.

a:=series(mul(mul(mul(1/(1-x^(i*j*k))^(1/(i*j*k)),k=1..21),j=1..50),i=1..50),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Apr 02 2019

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

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(tau_3(k)/k), where tau_3 = A007425.

E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} Sum_{j|d} tau(j) ) * x^k/k), where tau = number of divisors (A000005).

cp's OEIS Frontend

A321191 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^tau_n(k), where tau_n(k) = number of ordered n-factorizations of k.

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A318966 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} 1/(1 - x^(ijk))^(1/(ijk)).

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cp's OEIS Frontend

A321191 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^tau_n(k), where tau_n(k) = number of ordered n-factorizations of k.

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Author

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Mathematica

Formula

A318966 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} 1/(1 - x^(i*j*k))^(1/(i*j*k)).

Views

Author

Keywords

Links

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Programs

Maple

Mathematica

Formula

A318966 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} 1/(1 - x^(ijk))^(1/(ijk)).