A130898 -id:A130898 - cp's OEIS frontend

A130900 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.

1, 2, 3, 5, 6, 10, 12, 17, 22, 29, 36, 48, 58, 73, 91, 111, 134, 165, 197, 236, 283, 335, 395, 468, 547, 639, 747, 866, 1001, 1160, 1334, 1530, 1757, 2007, 2286, 2606, 2958, 3349, 3793, 4281, 4821, 5430, 6097, 6833, 7657, 8559, 9549, 10652, 11858, 13178

Offset: 1

Graeme McRae, Jun 07 2007

nonn

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.

			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.

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

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.

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@@5)(n->n):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 13 2011

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 *)

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

Original entry on oeis.org

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

Graeme McRae, Jun 07 2007

nonn

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.

			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.

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

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.

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

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 *)

cp's OEIS Frontend

A130900 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.

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