A141799 Number of repeated integer partitions of n.

1, 3, 8, 25, 66, 192, 511, 1418, 3812, 10383, 27958, 75758, 204215, 551821, 1488561, 4018722, 10842422, 29262357, 78955472, 213063551, 574905487, 1551325859, 4185959285, 11295211039, 30478118079, 82240300045, 221911189754, 598790247900, 1615732588962

Offset: 1

Author

Thomas Wieder, Jul 05 2008

Keywords

• nonn

Comments

An integer n can be partitioned into P(n) partitions P([n],i) where i=1,...,P(n) counts the partitions. The partition P([n],i) consists of T(n,i) integer parts t(i,j) with j=1,...,T(n,i). Now we perform on each t(i,j) an integer partition again and arrive at new partitions. Their parts can be partitioned again and so forth. We count such repeated partitions of n. One convention is necessary to avoid an infinite loop: The trivial partition P([n],1)=[n] will not be partitioned again but just counted once (and therefore we also have a(1)=1).

Examples

			For the integers 1, 2, 3 and 4 we have
[1] -> 1,
thus a(1)=1.
[2] -> 1,
[1,1] => [1] ->, [1] -> 1.
thus a(2)=3.
[3] -> 1,
[1,2] => [1] -> 1, [2] -> 3,
[1,1,1] => [1] -> 1, [1] -> 1, [1] -> 1,
thus a(3)=8.
[4] -> 1,
[1,3] => [1] -> 1, [3] -> 8,
[2,2] => [2] -> 3, [2] -> 3,
[1,1,2] => [1] -> 1, [1] -> 1, [2] -> 3,
[1,1,1,1] => [1] -> 1, [1] -> 1, [1] -> 1, [1] -> 1,
thus a(4)=25.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000041, A131407, A131408, A137732, A246828.

Programs

Maple

A141799 := proc(n) option remember ; local a,P,i,p ; if n =1 then 1; else a := 0 ; for P in combinat[partition](n) do if nops(P) > 1 then for i in P do a := a+procname(i) ; od: else a := a+1 ; fi; od: RETURN(a) ; fi ; end: for n from 1 to 40 do printf("%d,",A141799(n)) ; od: # R. J. Mathar, Aug 25 2008
# second Maple program
a:= proc(n) option remember;
      1+ `if`(n>1, b(n, n-1)[2], 0)
    end:
b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2] +g[1]*a(i)]
      fi
    end:
seq(a(n), n=1..40); # Alois P. Heinz, Apr 05 2012

Mathematica

a[n_] := a[n] = 1 + If[n>1, b[n, n-1][[2]], 0]; b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0 || i == 1, {1, n}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]*a[i]}]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Oct 29 2015, after Alois P. Heinz *)

Formula

Let sum_{i=1}^P(n) denote the sum over all integer partitions P([n],i) of n. Let sum_{j=1}^T(i,j) denote the sum over all parts of the i-th integer partition. Then we have the recursive formula 1 if t(i,j)=n a(n) = sum_{i=1}^P(n) sum_{j=1}^T(i,j) { a(t(i,j)) else. E.g. a(4)=25 because [4] contributes 1, [1,3] contributes a(1)+a(3)=1+8=9, [2,2] contributes a(2)+a(2)=3+3=6, [1,1,2] contributes a(1)+a(1)+a(2)=1+1+3=5, [1,1,1,1] contributes a(1)+a(1)+a(1)+a(1)=1+1+1+1=4 which gives in total 25.

a(n) ~ c * d^n, where d = 2.69832910647421123126399866... (see A246828), c = 0.5088820425072641934222229579416714164592334575899644931509447692360546... . - Vaclav Kotesovec, Sep 04 2014

Extensions

Extended by R. J. Mathar, Aug 25 2008