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

%I A130898 #10 Oct 26 2015 10:34:23
%S A130898 1,2,3,5,6,10,12,18,22,30,37,50,59,78,93,118,140,176,206,255,297,362,
%T A130898 421,507,585,699,803,949,1088,1276,1455,1696,1927,2230,2527,2909,3284,
%U A130898 3761,4233,4825,5416,6146,6879,7778,8682,9778,10892,12226,13582,15200
%N A130898 Number of partitions of n into "number of partitions of n into partition numbers" numbers.
%C A130898 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.
%C A130898 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.
%H A130898 Alois P. Heinz, <a href="/A130898/b130898.txt">Table of n, a(n) for n = 1..1000</a>
%e A130898 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.
%p A130898 pp:= proc(p) local b;
%p A130898        b:= proc(n, i)
%p A130898              if n<0 then 0
%p A130898            elif n=0 then 1
%p A130898            elif i<1 then 0
%p A130898            else b(n,i):= b(n,i-1) +b(n-p(i), i)
%p A130898              fi
%p A130898            end;
%p A130898        n-> b(n, n)
%p A130898      end:
%p A130898 a:= (pp@@3)(n->n):
%p A130898 seq(a(n), n=1..100); # _Alois P. Heinz_, Sep 13 2011
%t A130898 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_ *)
%Y A130898 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.
%K A130898 nonn
%O A130898 1,2
%A A130898 _Graeme McRae_, Jun 07 2007