A130899 Number of partitions of n into "number of partitions of n into 'number of partitions of n into partition numbers' numbers" numbers.

1, 2, 3, 4, 6, 9, 11, 15, 19, 25, 31, 41, 49, 61, 75, 91, 109, 134, 156, 188, 221, 262, 305, 361, 416, 485, 560, 648, 740, 858, 972, 1115, 1266, 1441, 1627, 1851, 2078, 2348, 2634, 2965, 3309, 3721, 4138, 4625, 5143, 5728, 6344, 7059, 7792, 8637, 9525, 10529

Offset

1,2

Comments

The "partition transformation" of sequence A can be defined as the number of partitions of n into elements of sequence A. This sequence (A130899) is the partition transformation composed with itself four times on the positive integers.

Links

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

Example

a(6) = 9 because there are 9 partitions of 6 whose parts are 1,2,3,5,6 which are terms of sequence A130898, which is the number of partitions of n into numbers of partitions of n into partition numbers.

Maple

pp:= proc(p) local b;
       b:= proc(n, i)
             if n<0 then 0
           elif n=0 then 1
           elif i<1 then 0
           else b(n, i):= b(n, i-1) +b(n-p(i), i)
             fi
           end;
       n-> b(n, n)
     end:
a:= (pp@@4)(n->n):
seq(a(n), n=1..100); # Alois P. Heinz, Sep 13 2011

Mathematica

pp[p_] := Module[{b}, b[n_, i_] := Which[n < 0, 0, n == 0, 1, i < 1, 0, True, b[n, i] = b[n, i-1] + b[n-p[i], i]]; b[#, #]&]; a = Nest[pp, Identity, 4]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

Crossrefs

Cf. A000027, A000041, A007279, A130898, A130900 which are m-fold self-compositions of the "partition transformation" on the counting numbers, for m=0, 1, 2, 4, 5.

Sequence in context: A304428 A132134 A027594 * A007210 A198394 A035947

Adjacent sequences: A130896 A130897 A130898 * A130900 A130901 A130902

Keyword

nonn

Author

Graeme McRae, Jun 07 2007

Status

approved