A304965 -id:A304965 - cp's OEIS frontend

A321287 Expansion of Product_{k>=1} (1 + x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

1, 1, 2, 5, 14, 22, 70, 109, 318, 551, 1203, 2136, 5752, 9263, 20641, 37151, 85084, 144918, 317356, 546730, 1196302, 2076810, 4281584, 7459351, 15860805, 27146911, 54715933, 95712097, 194059563, 334322338, 663159101, 1147479053, 2270647257, 3923732160, 7587368893

Offset: 0

Seiichi Manyama, Nov 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A163767, A304965, A321192.

tau[n_,1] = 1; tau[n_,k_]:=tau[n,k] = Plus @@ (tau[#, k-1] & /@ Divisors[n]); nmax = 40; CoefficientList[Series[Product[(1+x^k)^tau[k,k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 03 2018, after Robert G. Wilson v *)

A305049 Expansion of 1/(1 - Sum_{k>=1} tau_k(k)*x^k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

Original entry on oeis.org

1, 1, 3, 8, 27, 67, 216, 569, 1747, 4812, 14041, 39483, 115408, 326385, 941735, 2684170, 7725097, 22063737, 63354066, 181223899, 519883185, 1488316952, 4266788191, 12219763777, 35023995792, 100326757107, 287503501905, 823654031283, 2360146144917, 6761847714698, 19374935267810

Offset: 0

Ilya Gutkovskiy, May 24 2018

nonn

Invert transform of A163767.

N. J. A. Sloane, Transforms
Index entries for sequences related to compositions

Cf. A001906, A055887, A088305, A129373, A129374, A129921, A163767, A304963, A304964, A304965.

A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(A(j$2)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, May 24 2018

nmax = 30; CoefficientList[Series[1/(1 - Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]

G.f.: 1/(1 - Sum_{k>=1} A163767(k)*x^k).

cp's OEIS Frontend

A321287 Expansion of Product_{k>=1} (1 + x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

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