A130898 Number of partitions of n into "number of partitions of n into partition numbers" numbers.

1, 2, 3, 5, 6, 10, 12, 18, 22, 30, 37, 50, 59, 78, 93, 118, 140, 176, 206, 255, 297, 362, 421, 507, 585, 699, 803, 949, 1088, 1276, 1455, 1696, 1927, 2230, 2527, 2909, 3284, 3761, 4233, 4825, 5416, 6146, 6879, 7778, 8682, 9778, 10892, 12226, 13582, 15200

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 is the partition transformation composed with itself three times on the positive integers.

a(6) = 10 because there are 10 partitions of 6 whose parts are 1,2,3,4,6 which are terms of sequence A007279, which is the number of partitions of n into partition numbers.

Links

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

Example

a(6) = 12 because there are 12 partitions of 6 whose parts are 1,2,3,4,6 which are terms of sequence A007279, which is the number 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@@3)(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, 3]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

Crossrefs

Cf. A000027, A000041, A007279, A130899, 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: A115029 A241443 A023025 * A199016 A088314 A304405

Adjacent sequences: A130895 A130896 A130897 * A130899 A130900 A130901

Keyword

nonn

Author

Graeme McRae, Jun 07 2007

Status

approved