A130900 -id:A130900 - cp's OEIS frontend

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

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 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.

			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.

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

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.

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

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 *)

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 *)

A007211 Oscillates under partition transform.

Original entry on oeis.org

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, 14639

Offset: 1

N. J. A. Sloane, Mira Bernstein

Appears to be identical to A130900.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A007210, A027593, A130900

More terms from Sean A. Irvine, Nov 10 2019

cp's OEIS Frontend

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

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

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A007211 Oscillates under partition transform.

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