A130900 - OEIS

%I #10 Oct 26 2015 10:32:12

%S 1,2,3,5,6,10,12,17,22,29,36,48,58,73,91,111,134,165,197,236,283,335,

%T 395,468,547,639,747,866,1001,1160,1334,1530,1757,2007,2286,2606,2958,

%U 3349,3793,4281,4821,5430,6097,6833,7657,8559,9549,10652,11858,13178

%N Number of partitions of n into {number of partitions of n into "number of partitions of n into 'number of partitions of n into partition numbers' numbers" numbers} numbers.

%C The "partition transformation" of sequence A can be defined as the number of partitions of n into elements of sequence A. This sequence (A130900) is the partition transformation composed with itself five times on the positive integers.

%H Alois P. Heinz, <a href="/A130900/b130900.txt">Table of n, a(n) for n = 1..1000</a>

%e a(6) = 10 because there are 10 partitions of 6 whose parts are 1,2,3,4,6 which are terms of sequence A130899, which is the number of partitions of n into numbers of partitions of n into numbers of partitions of n into partition numbers.

%p pp:= proc(p) local b;

%p        b:= proc(n, i)

%p              if n<0 then 0

%p            elif n=0 then 1

%p            elif i<1 then 0

%p            else b(n,i):= b(n,i-1) +b(n-p(i), i)

%p              fi

%p            end;

%p        n-> b(n, n)

%p      end:

%p a:= (pp@@5)(n->n):

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Sep 13 2011

%t 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, 5]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Oct 26 2015, after _Alois P. Heinz_ *)

%Y Cf. A000027, A000041, A007279, A130898, A130899, A130900 which are m-fold self-compositions of the "partition transformation" on the counting numbers, m=0, 1, 2, 4, 5.

%K nonn

%O 1,2

%A _Graeme McRae_, Jun 07 2007