A304965 Expansion of Product_{k>=1} 1/(1 - x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

1, 1, 3, 6, 19, 30, 96, 152, 461, 775, 1883, 3271, 8751, 14370, 34004, 59491, 140450, 239746, 541817, 932681, 2089189, 3606641, 7719178, 13398411, 28848808, 49603982, 103047935, 179154858, 370200348, 639269735, 1295389370, 2241994088, 4511677298, 7798101800, 15408901600

Offset

0,3

Comments

Euler transform of A163767.

Links

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

N. J. A. Sloane, Transforms

Index entries for sequences related to partitions

Formula

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

Maple

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(add(d*
      A(d$2), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2018

Mathematica

nmax = 34; CoefficientList[Series[Product[1/(1 - x^k)^Times@@(Binomial[# + k - 1, k - 1]&/@FactorInteger[k][[All, 2]]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Times@@(Binomial[# + d - 1, d - 1]&/@FactorInteger[d][[All, 2]]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]

Crossrefs

Cf. A000219, A001970, A006171, A129373, A129374, A163767, A174465, A280487.

Sequence in context: A306968 A090956 A108972 * A203797 A019097 A347685

Adjacent sequences: A304962 A304963 A304964 * A304966 A304967 A304968

Keyword

nonn

Author

Ilya Gutkovskiy, May 22 2018

Status

approved