

A027293


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


26



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
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OFFSET

1,4


COMMENTS

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 nk, 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(k1))_{k>=0}.  Boris Putievskiy, Dec 14 2012


LINKS

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.


FORMULA

P(n,k) = p(nk) = A000041(nk), 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*n7))/2).  Boris Putievskiy, Dec 14 2012
From Wolfdieter Lang, Apr 14 2021: (Start)
Pa(n, m) = P(n+1, m+1) = A000041(nm), 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 'Leigensequence' B = A067687, that is, Pa*vec(B) = L*vec(B), with the matrix L with elements L(i, j) = delta(i, j1) (Kronecker's delta symbol). For such Leigensequences see the Bernstein and Sloane links under A155002.
Thanks to Gary W. Adamson for motivating me to look at such matrices and sequences. (End)


EXAMPLE

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


MATHEMATICA

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


CROSSREFS

Every column of P is A000041.
Cf. A145975, A010815.
Cf. A000070, A146023.
Cf. A182700, A027293, A067687, A155002.
Cf. A343234 (Leigenmatrix).
Sequence in context: A345418 A194543 A287920 * A104762 A152462 A180360
Adjacent sequences: A027290 A027291 A027292 * A027294 A027295 A027296


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling


STATUS

approved



