A063834 Twice partitioned numbers: the number of ways a number can be partitioned into not necessarily different parts and each part is again so partitioned.

1, 1, 3, 6, 15, 28, 66, 122, 266, 503, 1027, 1913, 3874, 7099, 13799, 25501, 48508, 88295, 165942, 299649, 554545, 997281, 1817984, 3245430, 5875438, 10410768, 18635587, 32885735, 58399350, 102381103, 180634057, 314957425, 551857780, 958031826, 1667918758

Offset: 0

Author

Wouter Meeussen, Aug 21 2001

Keywords

• nonn
• nice
• changed

Comments

These are different from plane partitions.

For ordered partitions of partitions see A055887 which may be computed from A036036 and A048996. - Alford Arnold, May 19 2006

Twice partitioned numbers correspond to triangles (or compositions) in the multiorder of integer partitions. - Gus Wiseman, Oct 28 2015

Examples

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + 266*x^8 + ...
If n=6, a possible first partitioning is (3+3), resulting in the following second partitionings: ((3),(3)), ((3),(2+1)), ((3),(1+1+1)), ((2+1),(3)), ((2+1),(2+1)), ((2+1),(1+1+1)), ((1+1+1),(3)), ((1+1+1),(2+1)), ((1+1+1),(1+1+1)).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000 (terms n=1..500 from T. D. Noe)
• Vaclav Kotesovec, Screenshot - A closed form formula for the constant c
• Gus Wiseman, Comcategories and Multiorders
• Gus Wiseman, Sequences enumerating triangles of integer partitions
• Gus Wiseman, On the Categorical Structure of Weakly Ordered Sequences of Integer Partitions

Crossrefs

Cf. A063835, A196545.

Cf. A036036, A048996, A055887.

Cf. A006906, A270995.

Cf. A007425, A047966, A047968, A271619, A279375, A279784-A279791.

The strict case is A296122.

Row sums of A321449.

Column k=2 of A323718.

Without singletons we have A327769, A358828, A358829.

For odd lengths we have A358823, A358824.

For distinct lengths we have A358830, A358912.

For strict partitions see A358914, A382524.

A000041 counts integer partitions, strict A000009.

A001970 counts multiset partitions of integer partitions.

Cf. A075900, A358831, A358833, A358906, A358908, A358909.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1)+`if`(i>n, 0, numbpart(i)*b(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 26 2015

Mathematica

Table[Plus @@ Apply[Times, IntegerPartitions[i] /. i_Integer :> PartitionsP[i], 2], {i, 36}]
(* second program: *)
b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i > n, 0, PartitionsP[i]*b[n-i, i]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)

PARI

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Dec 19 2016 */

Formula

G.f.: 1/Product_{k>0} (1-A000041(k)*x^k). n*a(n) = Sum_{k=1..n} b(k)*a(n-k), a(0) = 1, where b(k) = Sum_{d|k} d*A000041(d)^(k/d) = 1, 5, 10, 29, 36, 110, 106, ... . - Vladeta Jovovic, Jun 19 2003

From Vaclav Kotesovec, Mar 27 2016: (Start)

a(n) ~ c * 5^(n/4), where

c = 96146522937.7161898848278970039269600938032826... if n mod 4 = 0

c = 96146521894.9433858914667933636782092683849082... if n mod 4 = 1

c = 96146522937.2138934755566928890704687838407524... if n mod 4 = 2

c = 96146521894.8218716328341714149619262713426755... if n mod 4 = 3

(End)

Extensions

a(0)=1 prepended by Alois P. Heinz, Nov 26 2015