A343234 -id:A343234 - cp's OEIS frontend

A027293 Triangular array given by rows: P(n,k) is the number of partitions of n that contain k as a part.

1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 3, 2, 1, 1, 7, 5, 3, 2, 1, 1, 11, 7, 5, 3, 2, 1, 1, 15, 11, 7, 5, 3, 2, 1, 1, 22, 15, 11, 7, 5, 3, 2, 1, 1, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 77

Offset: 1

Clark Kimberling

Triangle read by rows in which row n lists the first n partition numbers A000041 in decreasing order. - Omar E. Pol, Aug 06 2011
A027293 * an infinite lower triangular matrix with A010815 (1, -1, -1, 0, 0, 1, ...) as the main diagonal the rest zeros = triangle A145975 having row sums = [1, 0, 0, 0, ...]. These matrix operations are equivalent to the comment in A010815 stating "when convolved with the partition numbers = [1, 0, 0, 0, ...]. - Gary W. Adamson, Oct 25 2008
From Gary W. Adamson, Oct 26 2008: (Start)
Row sums = A000070: (1, 2, 4, 7, 12, 19, 30, 45, 67, ...);
(this triangle)^2 = triangle A146023. (End)
(1) It appears that P(n,k) is also the total number of occurrences of k in the last k sections of the set of partitions of n (cf. A182703). (2) It appears that P(n,k) is also the difference, between n and n-k, of the total number of occurrences of k in all their partitions (cf. A066633). - Omar E. Pol, Feb 07 2012
Sequence B is called a reverse reluctant sequence of sequence A, if B is a triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. The present sequence is the reverse reluctant sequence of (A000041(k-1)){k>=0}. - _Boris Putievskiy, Dec 14 2012

			The triangle P begins (with offsets 0 it is Pa):
n \ k  1  2  3  4  5  6  7  8  9 10 ...
1:     1
2:     1  1
3:     2  1  1
4:     3  2  1  1
5:     5  3  2  1  1
6:     7  5  3  2  1  1
7:    11  7  5  3  2  1  1
8:    15 11  7  5  3  2  1  1
9:    22 15 11  7  5  3  2  1  1
10:   30 22 15 11  7  5  3  2  1  1
... reformatted by _Wolfdieter Lang_, Apr 14 2021

Robert Price, Table of n, a(n) for n = 1..5050
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Every column of P is A000041.
Cf. A145975, A010815.
Cf. A000070, A146023.
Cf. A182700, A027293, A067687, A155002.
Cf. A343234 (L-eigen-matrix).

f[n_] := Block[{t = Flatten[Union /@ IntegerPartitions@n]}, Table[Count[t, i], {i, n}]]; Array[f, 13] // Flatten
t[n_, k_] := PartitionsP[n-k]; Table[t[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 24 2014 *)

P(n,k) = p(n-k) = A000041(n-k), n>=1, k>=1. - Omar E. Pol, Feb 15 2013
a(n) = A000041(m), where m = (t*t + 3*t + 4)/2 - n, t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 14 2012
From Wolfdieter Lang, Apr 14 2021: (Start)
Pa(n, m) = P(n+1, m+1) = A000041(n-m), for n >= m >= 0, and 0 otherwise, gives the Riordan matrix Pa = (P(x), x), of Toeplitz type, with the o.g.f. P(x) of A000041. The o.g.f. of triangle Pa (the o.g.f. of the row polynomials RPa(n, x) = Sum_{m=0..n} Pa(n, m)*x^m) is G(z, x) = P(z)/(1 - x*z).
The (infinite) matrix Pa has the 'L-eigen-sequence' B = A067687, that is, Pa*vec(B) = L*vec(B), with the matrix L with elements L(i, j) = delta(i, j-1) (Kronecker's delta symbol). For such L-eigen-sequences see the Bernstein and Sloane links under A155002.
Thanks to Gary W. Adamson for motivating me to look at such matrices and sequences. (End)

A143866 Eigentriangle of A027293.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 5, 5, 3, 4, 5, 12, 7, 5, 6, 10, 12, 29, 11, 7, 10, 15, 24, 29, 69, 15, 11, 14, 25, 36, 58, 69, 165, 22, 15, 22, 35, 60, 87, 138, 165, 393, 937, 42, 30, 44, 75, 132, 203, 345, 495, 786, 937, 2233, 56, 42, 60, 110, 180, 319, 483, 825, 1179, 1874, 2233, 5322

Offset: 1

Gary W. Adamson, Sep 04 2008

Left border = partition numbers, A000041 starting (1, 1, 2, 3, 5, 7, ...). Right border = INVERT transform of partition numbers starting (1, 1, 2, 5, 12, ...); with row sums the same sequence but starting (1, 2, 5, 12, ...). Sum of n-th row terms = rightmost term of next row.

For another definition of L-eigen-matrix of A027293 see A343234. - Wolfdieter Lang, Apr 16 2021

			The triangle begins:
n \ k    1  2  3  4   5   6   7   8   9  10   11 ...
-------------------------------------------
1:       1
2:       1  1
3:       2  1  2
4:       3  2  2  5
5:       5  3  4  5  12
6:       7  5  6 10  12  29
7:      11  7 10 15  24  29  69
8:      15 11 14 25  36  58  69 165
9:      22 15 22 35  60  87 138 165 393
10:     30 22 30 55  84 145 207 330 393 937
11:     42 30 44 75 132 203 345 495 786 937 2233
... reformatted and extended by _Wolfdieter Lang_, May 02 2021
Row 4 = (3, 2, 2, 5) = termwise product of (3, 2, 1, 1) and (1, 1, 2, 5) = (3*1, 2*1, 1*2, 1*5).

Cf. A027293, A067687, A000041, A343234.

Triangle read by rows, A027293 * (A067687 * 0^(n-k)); 1 <= k <= n. (A067687 * 0^(n-k)) = an infinite lower triangular matrix with the INVERT transform of the partition function as the main diagonal: (1, 1, 2, 5, 12, 29, 69, 165, ...); and the rest zeros. Triangle A027293 = n terms of "partition numbers decrescendo"; by rows = termwise product of n terms of partition decrescendo and n terms of A027293: (1, 1, 2, 5, 12, 29, 69, 165, ...).

cp's OEIS Frontend

A027293 Triangular array given by rows: P(n,k) is the number of partitions of n that contain k as a part.

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