

A027293


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


25



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,...)
A027293^2 = 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 triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. Sequence A027293 is the reverse reluctant sequence of A000041.  Boris Putievskiy, Dec 14 2012


LINKS

Table of n, a(n) for n=1..79.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO]


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)/2n, t=floor((1+sqrt(8*n7))/2).  Boris Putievskiy, Dec 14 2012


EXAMPLE

Triangle begins:
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


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.
Sequence in context: A225640 A194543 A287920 * A104762 A152462 A180360
Adjacent sequences: A027290 A027291 A027292 * A027294 A027295 A027296


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling


STATUS

approved



