A146023 -id:A146023 - cp's OEIS frontend

A146024 Triangle read by rows, A146023 convolved with A010815.

1, 2, -1, 5, -2, -1, 10, -5, -2, 0, 20, -10, -5, 0, 0, 36, -20, -10, 0, 0, 1, 65, -36, -20, 0, 0, 2, 0, 1, 1, 0, 110, -65, -36, 0, 0, 5, 0, 1, 185, -110, -65, 0, 0, 110, 0, 2, 0, 300, -185, -110, 0, 0, 20, 0, 5, 0, 0

Offset: 0

Gary W. Adamson, Oct 26 2008

Row sums = the partition numbers, A000041: (1, 1, 2, 3, 5, 7, 11,...).

			First few rows of the triangle =
1;
2, -1;
5, -2, -1;
10, -5, -2, 0;
20, -10, -5, 0, 0;
36, -20, -10, 0, 0, 1;
65, -36, -20, 0, 0, 2, 0;
110, -65, -36, 0, 0, 5, 0, 1;
...

A146023, Cf. A000712, A000041

Triangle read by rows, A146023 * (A010815 * 0^(n-k)); 0<=k<=1.
Triangle A146023 has A000712 in every column.
(A010815 * 0^(n-k)) = an infinite lower triangular mtrix with A010815 as the right border and the rest zeros.
The operation A146023 * (A010815 * 0^(n-k)) convolves sequence A000712 with A010815.

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

Original entry on oeis.org

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)

A000713 EULER transform of 3, 2, 2, 2, 2, 2, 2, 2, ...

Original entry on oeis.org

1, 3, 8, 18, 38, 74, 139, 249, 434, 734, 1215, 1967, 3132, 4902, 7567, 11523, 17345, 25815, 38045, 55535, 80377, 115379, 164389, 232539, 326774, 456286, 633373, 874213, 1200228, 1639418, 2228546, 3015360, 4062065, 5448995, 7280060, 9688718, 12846507, 16972577

Offset: 0

N. J. A. Sloane

nonn

Equals row sums of triangle A146023. - Gary W. Adamson, Oct 26 2008
Partial sums of A000712. - Geoffrey Critzer, Apr 19 2012, corrected by Omar E. Pol, Jun 19 2012
Equals the number of partitions of n with 1's of three kinds and all parts >1 of two kinds. - Gregory L. Simay, Mar 25 2018

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 390
N. J. A. Sloane, Transforms

Cf. A000041, A000716.
Row sums of triangle A093010.
Cf. A146023. - Gary W. Adamson, Oct 26 2008

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> `if`(n<2,3,2)): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008

nn=20; g=Product[1/(1-x^i), {i,1,nn}]; c=1/(1-x); CoefficientList[Series[g^2/(1-x), {x,0,nn}], x] (* Geoffrey Critzer, Apr 19 2012 *)

x='x+O('x^66); Vec(1/((1-x)*eta(x)^2)) \\ Joerg Arndt, May 01 2013

from functools import lru_cache
from sympy import divisor_sigma
@lru_cache(maxsize=None)
def A000713(n): return sum(A000713(k)*((divisor_sigma(n-k)<<1)+1) for k in range(n))//n if n else 1 # Chai Wah Wu, Sep 25 2023

G.f.: A(x)/(1-x) where A(x) is g.f. for A000712. - Geoffrey Critzer, Apr 19 2012.
From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) ~ sqrt(3*n)/Pi * A000712(n).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*Pi*3^(1/4)*n^(3/4)).
(End)
G.f.: exp(Sum_{k>=1} (2*sigma_1(k) + 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018

Extended with formula from Christian G. Bower, Apr 15 1998

Definition changed by N. J. A. Sloane, Aug 15 2006

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A146024 Triangle read by rows, A146023 convolved with A010815.

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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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A000713 EULER transform of 3, 2, 2, 2, 2, 2, 2, 2, ...

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