A063835 Three times partitioned numbers: the number of ways a number can be partitioned in (not necessarily different) parts and each part again so partitioned a second and a third time.

1, 5, 14, 51, 125, 429, 1039, 3258, 8254, 23554, 58934, 168803, 412177, 1114550, 2795446, 7345875, 18035424, 46875324, 114272057, 291692396, 709742614, 1774402071, 4290848175, 10672950659, 25572179792, 62670553073, 149978278320

Offset: 1

Author

Wouter Meeussen, Aug 21 2001

Keywords

• nonn
• nice

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A063834.

Programs

Mathematica

Table[Plus@@((Apply[Plus, #/. i_Integer-> PartitionsP[i], {1}]/. f->Times)& /@ Flatten[Flatten[Outer[f, Sequence@@(Partitions/@#), 1]]&/@Partitions[w]]), {w, 16}]
nmax = 40; A047968 = Table[Sum[PartitionsP[d], {d, Divisors[n]}], {n, 1, nmax}]; conv = Table[Sum[A047968[[j]]*PartitionsP[m - j], {j, 1, m}], {m, 1, nmax}]; A063835 = Rest[CoefficientList[Series[Product[1/(1 - conv[[k]]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Mar 27 2016 *)

Formula

G.f.: 1/Product(1-b(n)*x^n, n=1..infinity), where b(n) is sum of number of partitions of parts in all partitions of n; b() is convolution of A047968() and A000041(). - Vladeta Jovovic, Nov 22 2005

From Vaclav Kotesovec, Mar 28 2016: (Start)

a(n) ~ c * 21^(n/4), where

c = 31506.382471540934704971753670563958673161001663... if mod(n,4) = 0

c = 31502.248225846169487427060315658509213347537914... if mod(n,4) = 1

c = 31506.175349116205868096360427802563935891182649... if mod(n,4) = 2

c = 31502.232274793501377850265964413938565498517297... if mod(n,4) = 3

(End)

Extensions

More terms from Vladeta Jovovic, Nov 22 2005