cp's OEIS Frontend

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

%I A130899 #10 Oct 26 2015 10:37:21
%S A130899 1,2,3,4,6,9,11,15,19,25,31,41,49,61,75,91,109,134,156,188,221,262,
%T A130899 305,361,416,485,560,648,740,858,972,1115,1266,1441,1627,1851,2078,
%U A130899 2348,2634,2965,3309,3721,4138,4625,5143,5728,6344,7059,7792,8637,9525,10529
%N A130899 Number of partitions of n into "number of partitions of n into 'number of partitions of n into partition numbers' numbers" numbers.
%C A130899 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.
%H A130899 Alois P. Heinz, <a href="/A130899/b130899.txt">Table of n, a(n) for n = 1..1000</a>
%e A130899 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.
%p A130899 pp:= proc(p) local b;
%p A130899        b:= proc(n, i)
%p A130899              if n<0 then 0
%p A130899            elif n=0 then 1
%p A130899            elif i<1 then 0
%p A130899            else b(n,i):= b(n,i-1) +b(n-p(i), i)
%p A130899              fi
%p A130899            end;
%p A130899        n-> b(n, n)
%p A130899      end:
%p A130899 a:= (pp@@4)(n->n):
%p A130899 seq(a(n), n=1..100); # _Alois P. Heinz_, Sep 13 2011
%t A130899 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_ *)
%Y A130899 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.
%K A130899 nonn
%O A130899 1,2
%A A130899 _Graeme McRae_, Jun 07 2007