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

1,2

Links

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

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)

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 *)

Crossrefs

Cf. A063834.

Sequence in context: A272223 A107242 A203164 * A054664 A091218 A197601

Adjacent sequences: A063832 A063833 A063834 * A063836 A063837 A063838

Keyword

nonn,nice

Author

Wouter Meeussen, Aug 21 2001

Extensions

More terms from Vladeta Jovovic, Nov 22 2005

Status

approved