A024790 -id:A024790 - cp's OEIS frontend

A141450 Upper right triangle of the number of m's in all partitions of n.

1, 1, 2, 1, 1, 4, 1, 1, 3, 7, 1, 1, 2, 4, 12, 1, 1, 2, 4, 8, 19, 1, 1, 2, 3, 6, 11, 30, 1, 1, 2, 3, 6, 9, 19, 45, 1, 1, 2, 3, 5, 8, 15, 26, 67, 1, 1, 2, 3, 5, 8, 13, 21, 41, 97, 1, 1, 2, 3, 5, 7, 12, 18, 31, 56, 139, 1, 1, 2, 3, 5, 7, 12, 17, 28, 45, 83, 195, 1, 1, 2, 3, 5, 7, 11, 16, 25, 38, 63

Offset: 1

Robert G. Wilson v, Aug 07 2008

The "last" column read from the bottom is A000041.

Mirror of triangle A066633. - Omar E. Pol, May 01 2012

			A000070: 1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 195, 272, 373, 508, ...,
A024786: 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 112, 160, 213, ...,
A024787: 0, 0, 1, 1, 2, 4, 6, 9, 15, 21, 31, 45, 63, 87, 122, ...,
A024788: 0, 0, 0, 1, 1, 2, 3, 6, 8, 13, 18, 28, 38, 55, 74, ...,
A024789: 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 50, ...,
A024790: 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 16, 24, 33, ...,
A024791: 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 16, 23, ...,
A024792: 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, ...,
A024793: 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, ...,
A024794: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, ...

Cf. A000041, A000070, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794.

(* First do ) Needs["Combinatorica`"] (* then *) f[n_, m_] := Count[Flatten@ Partitions@ n, m]; Table[ f[n, m], {n, 13}, {m, n, 1, -1}]

A188139 Triangle by rows, A027293 * A129372 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 4, 2, 1, 1, 8, 3, 2, 1, 1, 11, 6, 3, 2, 1, 1, 19, 8, 5, 3, 2, 1, 1, 26, 13, 7, 5, 3, 2, 1, 1, 41, 18, 12, 7, 5, 3, 2, 1, 1, 56, 28, 16, 11, 7, 5, 3, 2, 1, 1, 83, 38, 24, 15, 11, 7, 5, 3, 2, 1, 1, 112, 55, 33, 23, 15, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Gary W. Adamson, Mar 21 2011

Row sums = A066897: (1, 2, 5, 8, 15, 24, 39,...), total number of odd parts in all partitions of n.

Apparently T(n,k) is the number of (2*k)'s in all the partitions of (n+k), k>=1, e.g. T(7,3) = number of 6's in partitions of 10 = A024790(10). [David Scambler, May 24 2012]

			First few rows of the triangle =
.
1,
1, 1
3, 1, 1
4, 2, 1, 1
8, 3, 2, 1, 1
11, 6, 3, 2, 1, 1
19, 8, 5, 3, 2, 1, 1
26, 13, 7, 5, 3, 2, 1, 1
41, 18, 12, 7, 5, 3, 2, 1, 1
56, 28, 16, 11, 7, 5, 3, 2, 1, 1
83, 38, 24, 15, 11, 7, 5, 3, 2, 1, 1
112, 55, 33, 23, 15, 11, 7, 5, 3, 2, 1, 1
160, 74, 47, 31, 22, 15, 11, 7, 5, 3, 2, 1, 1,
...

Cf. A027293, A066897, A129372.

Table[Count[Flatten[IntegerPartitions[n+k]], 2*k], {n,1,15}, {k,1,n}] (* David Scambler, May 24 2012 *)

a(22) ff. corrected and more terms from Georg Fischer, Jun 10 2023

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A141450 Upper right triangle of the number of m's in all partitions of n.

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