A027293 -id:A027293 - cp's OEIS frontend

A145975 Triangle read by rows, partition triangle A027293 convolved with A010815.

1, 1, -1, 2, -1, -1, 3, -2, -1, 0, 5, -3, -2, 0, 0, 7, -5, -3, 0, 0, 1, 11, -7, -5, 0, 0, 1, 0, 15, -11, -7, 0, 0, 2, 0, 1, 22, -15, -11, 0, 0, 3, 0, 1, 0, 30, -22, -15, 0, 0, 5, 0, 2, 0, 0, 42, -30, -22, 0, 0, 7, 0, 3, 0, 0, 0, 56, -42, -30, 0, 0, 11, 0, 5, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Oct 25 2008

Row sums = [1, 0, 0, 0,...]. (a set of matrix operations equivalent to the comment in A010815 that "convolved with the partition numbers = [1, 0, 0, 0,...].

			First few rows of the triangle =
1;
1, -1;
2, -1, -1;
3, -2, -1, 0;
5, -3, -2, 0, 0;
7, -5, -3, 0, 0, 1;
11, -7, -5, 0, 0, 1, 0;
15, -11, -7, 0, 0, 2, 0, 1;
22, -15, -11, 0, 0, 3, 0, 1, 0;
30, -22, -15, 0, 0, 5, 0, 2, 0, 0;
42, -30, -22, 0, 0, 7, 0, 3, 0, 0, 0;
...

Robert Price, Table of n, a(n) for n = 1..1275 (first 50 rows)

A027293, Cf. A010815, A000041

Table[Count[Flatten[Union /@ IntegerPartitions@n],k]*SeriesCoefficient[Product[1 - x^i, {i, k - 1}], {x, 0, k - 1}], {n, 12}, {k, n}] // Flatten (* Robert Price, Jun 15 2020 *)

Triangle read by rows, = (A027293 * (A010815 * 0^(n-k)); 0<=k<=n. A027293 = an infinite lower triangular matrix with A000041 in every column (the partition numbers). A010815 = (1, -1, -1, 0, 0, 1,...)

Missing zero at a(55) inserted by Robert Price, Jun 15 2020

A146023 Triangle read by rows, square of A027293.

Original entry on oeis.org

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

Offset: 0

Gary W. Adamson, Oct 26 2008

Triangle, A000712 (1, 2, 5, 10, 20, 36,...) in every column.
Row sums = A000713: (1, 3, 8, 18, 38, 74, 139,...)
Note that we are working here in the world of lower triangular matrices. - N. J. A. Sloane, Jun 15 2020

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

Robert Price, Table of n, a(n) for n = 0..819 (first 40 rows)

Cf. A000712, A000713, A027293.

lim = 10;
A000712 = Table [Length@IntegerPartitions[n, All, Range@n~Join~Range@n], {n, 0, lim - 1}]
Table[Reverse[Take[A000712, n]], {n, lim}] // Flatten (* Robert Price, Jun 15 2020 *)

Extraneous data deleted by Robert Price, Jun 15 2020

A091114 Number of partitions of n-th composite number containing the smallest prime factor: a(n) = A027293(A002808(n), A056608(n)).

Original entry on oeis.org

2, 5, 11, 11, 22, 42, 77, 77, 135, 231, 385, 385, 627, 1002, 627, 1575, 1575, 2436, 3718, 5604, 5604, 8349, 5604, 12310, 17977, 17977, 26015, 37338, 53174, 53174, 75175, 105558, 53174, 147273, 147273, 204226, 281589, 204226, 386155, 386155

Offset: 1

Reinhard Zumkeller, Feb 22 2004

nonn

a(n) = A000041(A002808(n)) - A091094(n).

a(n) = A000041(A085271(n)). - Charlie Neder, Jan 10 2019

			n=2: A002808(2)=6=2*3 has A000041(6)=11 partitions: 6 = 5+1 = 4+2 = 4+1+1 = 3+3 = 3+2+1 = 3+1+1+1 = 2+2+2 = 2+2+1+1 = 2+1+1+1+1 = 1+1+1+1+1+1, 2 occurs in 5 partitions, therefore a(2)=5.

Cf. A002808, A000041, A020639, A027293, A056608, A085271, A091109, A091094.

lista(nn) = forcomposite(n=2, nn, print1(numbpart(n - divisors(n)[2]), ", ")); \\ Michel Marcus, Jan 11 2019

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, ...).

A152538 Triangle read by rows, A027293 * (A152537 * 0^(n-k)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 5, 3, 2, 2, 4, 7, 5, 3, 4, 4, 9, 11, 7, 5, 6, 8, 9, 18, 15, 11, 7, 10, 12, 18, 18, 37, 22, 15, 11, 14, 20, 27, 36, 37, 74, 30, 22, 15, 22, 28, 45, 54, 74, 74, 148, 42, 30, 22, 30, 44, 63, 90, 111, 148, 148, 296

Offset: 0

Gary W. Adamson, Dec 10 2008

Row sums = 2^n.

Right border = A152537, left border = A000041.

			First few rows of the triangle =
1;
1, 1;
2, 1, 1;
3, 2, 1, 2;
5, 3, 2, 2, 4;
7, 5, 3, 4, 4, 9;
11, 7, 5, 6, 8, 9, 18;
15, 11, 7, 10, 12, 18, 18, 37;
22, 15, 11, 14, 20, 27, 36, 37, 74;
30, 22, 15, 22, 28, 45, 54, 74, 74, 148;
42, 30, 22, 30, 44, 63, 90, 111, 148, 148, 296;
56, 42, 30, 44, 60, 99, 126, 185, 222, 296, 296, 592;
77, 56, 42, 60, 88, 135, 198, 259, 370, 444, 592, 592, 1183;
...
Row 3 = (3, 2, 1, 2) = termwise products of (3, 2, 1, 1) and (1, 1, 1, 2).

Cf. A152537, A027293, A000041

Triangle read by rows, M*Q. M = A027293 as an infinite lower triangular matrix with the partition numbers (A000041) in every column. Q = a matrix with A152537 as the main diagonal and the rest zeros.

A174740 Triangle read by rows, A027293 * an infinite lower triangular matrix with A147843 as the diagonal and the rest zeros.

Original entry on oeis.org

1, 1, 2, 2, 2, 0, 3, 4, 0, 0, 5, 6, 0, 0, -5, 7, 10, 0, 0, -5, 0, 11, 14, 0, 0, -10, 0, -7, 15, 22, 0, 0, -15, 0, -7, 0, 22, 30, 0, 0, -25, 0, -14, 0, 0, 30, 44, 0, 0, -35, 0, -21, 0, 0, 0, 42, 60, 0, 0, -55, 0, -35, 0, 0, 0, 0, 56, 84, 0, 0, -75, 0, -49, 0, 0, 0, 0, 12

Offset: 1

Gary W. Adamson, Mar 28 2010

Left border = the partition numbers, A000041; right border = A147843 starting (1, 2, 0, ...).

Row sums apparently give A000203. Check: Sum of row 6 terms = [7, 5, 3, 2, 1, 1] dot [1, 2, 0, 0, -5, 0] = [7 + 10 + 0 + 0 -5 + 0] = 12 = A000203(6).

			First few rows of the triangle:
   1;
   1,   2;
   2,   2,   0;
   3,   4,   0,   0;
   5,   6,   0,   0,  -5;
   7,  10,   0,   0,  -5,   0;
  11,  14,   0,   0, -10,   0,  -7;
  15,  22,   0,   0, -15,   0,  -7,   0;
  22,  30,   0,   0, -25,   0, -14,   0,   0;
  30,  44,   0,   0, -35,   0, -21,   0,   0,   0;
  42,  60,   0,   0, -55,   0, -35,   0,   0,   0,   0;
  56,  84,   6,   0, -75,   0, -49,   0,   0,   0,   0,  12;
  ...

Cf. A027293, A147843, A000041, A000203.

Equals triangle A027293 * a lower triangular matrix with A147843 (deleting the first zero) as the right border and the rest zeros.

T(n,k) = A147843(k) * A027293(n,k). - Joerg Arndt, Dec 29 2022

Terms corrected by Gary W. Adamson, Dec 27 2022

A147768 Triangle read by rows: A000012^(-2) * A027293 as infinite lower triangular matrices.

Original entry on oeis.org

1, -1, 1, 1, -1, 1, 0, 1, -1, 1, 1, 0, 1, -1, 1, 0, 1, 0, 1, -1, 1, 2, 0, 1, 0, 1, -1, 1, 0, 2, 0, 1, 0, 1, -1, 1, 3, 0, 2, 0, 1, 0, 1, -1, 1, 1, 3, 0, 2, 0, 1, 0, 1, -1, 1, 4, 1, 3, 0, 2, 0, 1, 0, 1, -1, 1, 2, 4, 1, 3, 0, 2, 0, 1, 0, 1, -1, 1

Offset: 0

Gary W. Adamson, Nov 11 2008

Row sums = A002865, (1, 0, 1, 1, 2, 2, 4, 4, 7, 8,...).

This triangle is the lower right half of a Toeplitz matrix. Each column of this triangle has the form [1, -1] U A053445. - Georg Fischer, Jul 28 2023

			First few rows of the triangle =
1;
-1, 1;
1, -1, 1;
0, 1, -1, 1;
1, 0, 1, -1, 1;
0, 1, 0, 1, -1, 1;
2, 0, 1, 0, 1, -1, 1;
0, 2, 0, 1, 0, 1, -1, 1;
3, 0, 2, 0, 1, 0, 1, -1, 1;
1, 3, 0, 2, 0, 1, 0, 1, -1, 1;
4, 1, 3, 0, 2, 0, 1, 0, 1, -1, 1;
2, 4, 1, 3, 0, 2, 0, 1, 0, 1, -1, 1;
...

Cf. A000012, A000041, A002865, A004736, A027293, A053445, A185018.

A000012^(-1) is the pairwise difference operator, and A027293 = a triangle with A000041 in every column.

Equals A185018 * A027293 since A000012^2 = A004736 and A004736^(-1) = A185018. - Georg Fischer, Jul 28 2023

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

A000070 a(n) = Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 195, 272, 373, 508, 684, 915, 1212, 1597, 2087, 2714, 3506, 4508, 5763, 7338, 9296, 11732, 14742, 18460, 23025, 28629, 35471, 43820, 53963, 66273, 81156, 99133, 120770, 146785, 177970, 215308, 259891, 313065, 376326, 451501

Offset: 0

N. J. A. Sloane

Also the total number of all different integers in all partitions of n + 1. E.g., a(3) = 7 because the partitions of 4 comprise the sets {1},{1, 2},{2},{1, 3},{4} of different integers and their total number is 7. - Thomas Wieder, Apr 10 2004
With offset 1, also the number of 1's in all partitions of n. For example, 3 = 2+1 = 1+1+1, a(3) = (zero 1's) + (one 1's) + (three 1's), so a(3) = 4. - Naohiro Nomoto, Jan 09 2002. See the Riordan reference p. 184, last formula, first term, for a proof based on Fine's identity given in Riordan, p. 182 (20).
Also, number of partitions of n into parts when there are two kinds of parts of size one.
Also number of graphical forest partitions of 2n+2.
a(n) = count 2 for each partition of n and 1 for each decrement. E.g., the partitions of 4 are 4 (2), 31 (3), 22 (2), 211 (3) and 1111 (2). 2 + 3 + 2 + 3 + 2 = 12. This is related to the Ferrers representation. We can see that taking the Ferrers diagram for each partition of n and adding a new * to all available columns, we generate each partition of n+1, but with repeats (A058884). - Jon Perry, Feb 06 2004
Also the number of 1-transitions among all integer partitions of n. A 1-transition is the removal of a digit "1" from a partition containing at least one "1" and subsequent addition of that "1" to another digit in that partition. This other digit may be a "1" also, but all digits of equal amount are considered as undistinquishable (unlabeled). E.g., for n=6 one has the partition [1113] for which the following two 1-transitions are possible: [1113] --> [123] and [1113] --> [114]. The 1-transitions of n form a partial order (poset). For n=6 one has 12 1-transitions: [111111] --> [11112], [11112] --> [1113], [11112] --> [1122], [1113] --> [114], [1113] --> [123], [1122] --> [123], [1122] --> [222], [123] --> [33], [123] --> [24], [114] --> [15], [114] --> [24], [15] --> [6]. - Thomas Wieder, Mar 08 2005
Also number of partitions of 2n+1 where one of the parts is greater than n (also where there are more than n parts) and of 2n+2 where one of the parts is greater than n+1 (or with more than n+1 parts). - Henry Bottomley, Aug 01 2005
Equals left border of triangle A137633 - Gary W. Adamson, Jan 31 2008
Equals row sums of triangle A027293. - Gary W. Adamson, Oct 26 2008
Convolved with A010815 = [1,1,1,...]. n-th partial sum of A000041 convolved with A010815 = the binomial sequence starting (1, n, ...). - Gary W. Adamson, Nov 09 2008
Equals A036469 convolved with A035363. - Gary W. Adamson, Jun 09 2009
a(A004526(n)) = A025065(n). - Reinhard Zumkeller, Jan 23 2010
a(n) = if n <= 1 then A054225(1,n) else A054225(n,1). - Reinhard Zumkeller, Nov 30 2011
Also the total number of 1's among all hook-lengths in all partitions of n. E.g., a(4)=7 because hooks of the partitions of n = 4 comprise the multisets {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1} and their total number of 1's is 7. - T. Amdeberhan, Jun 03 2012
With offset 1, a(n) is also the difference between the sum of largest and the sum of second largest elements in all partitions of n. More generally, the number of occurrences of k in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+1)st largest elements in all partitions of n. And more generally, the sum of the number of occurrences of k, k+1, k+2..k+m in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+m+1)st largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
a(0) = 1 and 2*a(n-1) >= a(n) for all n > 0. Hence a(n) is a complete sequence. - Frank M Jackson, Apr 08 2013
a(n) is the number of conjugacy classes in the order-preserving, order-decreasing and (order-preserving and order-decreasing) injective transformation semigroups. - Ugbene Ifeanyichukwu, Jun 03 2015
a(n) is also the number of unlabeled subgraphs of the n-cycle C_n. For example, for n = 3, there are 3 unlabeled subgraphs of the triangle C_3 with 0 edges, 2 with 1 edge, 1 with 2 edges, and 1 with 3 edges (C_3 itself), so a(3) = 3 + 2 + 1 + 1 = 7. - John P. McSorley, Nov 21 2016
a(n) is also the number of partitions of 2n with all parts either even or equal to 1. Proof: the number of such partitions of 2n with exactly 2k 1's is p(n-k), for k = 0,..,n. Summing over k gives the formula. - Leonard Chastkofsky, Jul 24 2018
a(n) is the total number of polygamma functions that appear in the expansion of the (n+1)st derivative of x! with respect to x. More specifically, a(n) is the number of times the string "PolyGamma" appears in the expansion of D[x!, {x, n + 1}] in Mathematica. For example,  D[x!, {x, 3 + 1}] = Gamma[1 + x] PolyGamma[0, 1 + x]^4 + 6 Gamma[1 + x] PolyGamma[0, 1 + x]^2 PolyGamma[1, 1 + x] + 3 Gamma[1 + x] PolyGamma[1, 1 + x]^2 + 4 Gamma[1 + x] PolyGamma[0, 1 + x] PolyGamma[2, 1 + x] + Gamma[1 + x] PolyGamma[3, 1 + x], and we see that the string "PolyGamma" appears a total of a(3) = 7 times in this expansion. - John M. Campbell, Aug 11 2018
With offset 1, also the number of integer partitions of 2n that do not comprise the multiset of vertex-degrees of any multigraph (i.e., non-multigraphical partitions); see A209816 for multigraphical partitions. - Gus Wiseman, Oct 26 2018
Also a(n) is the number of partitions of 2n+1 with exactly one odd part.
Delete the odd part 2k+1, k=0, ..., n, to get a partition of 2n-2k into even parts. There are as many unrestricted partitions of n-k; now sum those numbers from 0 to n to get a(n). - George Beck, Jul 22 2019
In the Young's lattice, a(n) is the number of branches that connect the (n-1)-th layer to the n-th layer. - Shouvik Datta, Sep 19 2021
a(n) is the number of multiset partitions of the multiset {r^n, s^1}, equivalently, factorization patterns of any number m=p^n*q^1 where p and q are primes. - Joerg Arndt, Jan 01 2024
a(n) is the number of positive integers whose divisors are the parts of the partitions of n + 1. - Omar E. Pol, Nov 07 2024

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 30*x^6 + 45*x^7 + 67*x^8 + ...
From _Omar E. Pol_, Oct 25 2012: (Start)
For n = 5 consider the partitions of n+1:
--------------------------------------
.                         Number
Partitions of 6           of 1's
--------------------------------------
6 .......................... 0
3 + 3 ...................... 0
4 + 2 ...................... 0
2 + 2 + 2 .................. 0
5 + 1 ...................... 1
3 + 2 + 1 .................. 1
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 2
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 4
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
35-16 =                     19
.
The difference between the sum of the first column and the sum of the second column of the set of partitions of 6 is 35 - 16 = 19 and equals the number of 1's in all partitions of 6, so the 6th term of this sequence is a(5) = 19.
(End)
From _Gus Wiseman_, Oct 26 2018: (Start)
With offset 1, the a(1) = 1 through a(6) = 19 partitions of 2*n whose greatest part is > n:
  (2)  (4)   (6)    (8)     (A)      (C)
       (31)  (42)   (53)    (64)     (75)
             (51)   (62)    (73)     (84)
             (411)  (71)    (82)     (93)
                    (521)   (91)     (A2)
                    (611)   (622)    (B1)
                    (5111)  (631)    (732)
                            (721)    (741)
                            (811)    (822)
                            (6211)   (831)
                            (7111)   (921)
                            (61111)  (A11)
                                     (7221)
                                     (7311)
                                     (8211)
                                     (9111)
                                     (72111)
                                     (81111)
                                     (711111)
With offset 1, the a(1) = 1 through a(6) = 19 partitions of 2*n whose number of parts is > n:
  (11)  (211)   (2211)    (22211)     (222211)      (2222211)
        (1111)  (3111)    (32111)     (322111)      (3222111)
                (21111)   (41111)     (331111)      (3321111)
                (111111)  (221111)    (421111)      (4221111)
                          (311111)    (511111)      (4311111)
                          (2111111)   (2221111)     (5211111)
                          (11111111)  (3211111)     (6111111)
                                      (4111111)     (22221111)
                                      (22111111)    (32211111)
                                      (31111111)    (33111111)
                                      (211111111)   (42111111)
                                      (1111111111)  (51111111)
                                                    (222111111)
                                                    (321111111)
                                                    (411111111)
                                                    (2211111111)
                                                    (3111111111)
                                                    (21111111111)
                                                    (111111111111)
(End)
From _Joerg Arndt_, Jan 01 2024: (Start)
The a(5) = 19 multiset partitions of the multiset {1^5, 2^1} are:
   1:  {{1, 1, 1, 1, 1, 2}}
   2:  {{1, 1, 1, 1, 1}, {2}}
   3:  {{1, 1, 1, 1, 2}, {1}}
   4:  {{1, 1, 1, 1}, {1, 2}}
   5:  {{1, 1, 1, 1}, {1}, {2}}
   6:  {{1, 1, 1, 2}, {1, 1}}
   7:  {{1, 1, 1, 2}, {1}, {1}}
   8:  {{1, 1, 1}, {1, 1, 2}}
   9:  {{1, 1, 1}, {1, 1}, {2}}
  10:  {{1, 1, 1}, {1, 2}, {1}}
  11:  {{1, 1, 1}, {1}, {1}, {2}}
  12:  {{1, 1, 2}, {1, 1}, {1}}
  13:  {{1, 1, 2}, {1}, {1}, {1}}
  14:  {{1, 1}, {1, 1}, {1, 2}}
  15:  {{1, 1}, {1, 1}, {1}, {2}}
  16:  {{1, 1}, {1, 2}, {1}, {1}}
  17:  {{1, 1}, {1}, {1}, {1}, {2}}
  18:  {{1, 2}, {1}, {1}, {1}, {1}}
  19:  {{1}, {1}, {1}, {1}, {1}, {2}}
(End)

H. Gupta, An asymptotic formula in partitions. J. Indian Math. Soc., (N. S.) 10 (1946), 73-76.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 6.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018
A. M. Odlyzko, Asymptotic Enumeration Methods, p. 19
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
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).
Stanley, R. P., Exercise 1.26 in Enumerative Combinatorics, Vol. 1. Cambridge, England: Cambridge University Press, p. 59, 1999.

T. D. Noe, Table of n, a(n) for n = 0..1000
P. A. Baikov and S. V. Mikhailov, The {beta}-expansion for Adler function, Bjorken Sum Rule, and the Crewther-Broadhurst-Kataev relation at order O(alpha_s^4), J. High Energy Phys. 09 (2022) Art. No. 185. See also arXiv:2206.14063 [hep-ph], 2022.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
Philip Boalch, Counting the fission trees and nonabelian Hodge graphs, arXiv:2410.23358 [math.AG], 2024. See p. 15.
L. Bracci and L. E. Picasso, A simple iterative method to write the terms of any order of perturbation theory in quantum mechanics, The European Physical Journal Plus, 127 (2012), Article 119. - From _N. J. A. Sloane_, Dec 31 2012
Emmanuel Briand, Samuel A. Lopes, and Mercedes Rosas, Normally ordered forms of powers of differential operators and their combinatorics, arXiv:1811.00857 [math.CO], 2018.
C. C. Cadogan, On partly ordered partitions of a positive integer, Fib. Quart., 9 (1971), 329-336.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers, National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956. (Annotated scanned pages from, plus a review)
Philip Cuthbertson, Fixed hooks in arbitrary columns of partitions, Integers (2025) Vol. 25, Art. No. A28. See p. 3.
Mario De Salvo, Dario Fasino, Domenico Freni and Giovanni Lo Faro, A Family of 0-Simple Semihypergroups Related to Sequence A000070, Journal of Multiple-Valued Logic & Soft Computing, 2016, Vol. 27, Issue 5/6, pp. 553-572.
Mario De Salvo, Dario Fasino, Domenico Freni, and Giovanni Lo Faro, Semihypergroups Obtained by Merging of 0-semigroups with Groups, Filomat 32(12) (2018), 4177-4194.
P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experimental Mathematics, 7(1) (1998), 15-35.
D. Frank, C. D. Savage and J. A. Sellers, On the Number of Graphical Forest Partitions, Ars Combinatoria, Vol. 65 (2002), 33-37.
D. Frank, C. D. Savage and J. A. Sellers, On the Number of Graphical Forest Partitions, preprint.
Manosij Ghosh Dastidar and Sourav Sen Gupta, Generalization of a few results in Integer Partitions, arXiv preprint arXiv:1111.0094 [cs.DM], 2011.
Petros Hadjicostas, Cyclic, Dihedral and Symmetrical Carlitz Compositions of a Positive Integer, Journal of Integer Sequences, 20 (2017), Article #17.8.5.
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
M. D. Hirschhorn, The number of 1's in the partitions of n, Fib. Quart., 51 (2013), 326-329.
M. D. Hirschhorn, The number of different parts in the partitions of n, Fib. Quart., 52 (2014), 10-15. See p. 11. - _N. J. A. Sloane_, Mar 25 2014
Nick Hobson, Solution to puzzle 56: Partition identity
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 113.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 126.
MathOverflow, Number of branches between two layers of the Young's lattice, Sep 19 2021.
Mikhailov, S. V. On a realization of beta-expansion in QCD, J. High Energy Phys. 2017, No. 4, Paper No. 169, 16 p. (2017).
M. M. Mogbonju, O. A. Ojo, and I. A. Ogunleke, Graphical Representation of Conjugacy Classes in the Order-Preserving Partial One-One Transformation Semigroup, International Journal of Science and Research (IJSR), 3(12) (2014), 711-721.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
F. Ruskey, Combinatorial Object Server.
Maria Schuld, Kamil Brádler, Robert Israel, Daiqin Su, and Brajesh Gupt, A quantum hardware-induced graph kernel based on Gaussian Boson Sampling, arXiv:1905.12646 [quant-ph], 2019.
N. J. A. Sloane, Transforms
I. J. Ugbene, E. O. Eze, and S. O. Makanjuola, On the Number of Conjugacy Classes in the Injective Order-Decreasing Transformation Semigroup, Pacific Journal of Science and Technology, 14(1) (2013), 182-186.
Ifeanyichukwu Jeff Ugbene, Gatta Naimat Bakare, and Garba Risqot Ibrahim, Conjugacy classes of the order-preserving and order-decreasing partial one-to-one transformation semigroups, Journal of Science, Technology, Mathematics and Education (JOSTMED), 15(2) (2019), 83-88.
Joseph Vandehey, Digital problems in the theory of partitions, Integers (2024) Vol. 24A, Art. No. A18. See p. 3.
Eric Weisstein's World of Mathematics, Stanley's Theorem.

A diagonal of A066633.
Also second column of A126442. - George Beck, May 07 2011
Row sums of triangle A092905.
Also row sums of triangle A261555. - Omar E. Pol, Sep 14 2016
Also row sums of triangle A278427. - John P. McSorley, Nov 25 2016
Column k=2 of A292508.
Cf. A014153, A024786, A026794, A026905, A058884, A093694, A133735, A137633, A010815, A027293, A035363, A028310, A000712, A000990.
Cf. A000569, A025065, A036469, A096373, A147878, A209816, A320891, A320924.

List([0..45],n->Sum([0..n],k->NrPartitions(k))); # Muniru A Asiru, Jul 25 2018

a000070 = p a028310_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 06 2012

with(combinat): a:=n->add(numbpart(j),j=0..n): seq(a(n), n=0..44); # Zerinvary Lajos, Aug 26 2008

CoefficientList[ Series[1/(1 - x)*Product[1/(1 - x^k), {k, 75}], {x, 0, 45}], x] (* Robert G. Wilson v, Jul 13 2004 *)
Table[ Count[ Flatten@ IntegerPartitions@ n, 1], {n, 45}] (* Robert G. Wilson v, Aug 06 2008 *)
Join[{1}, Accumulate[PartitionsP[Range[50]]]+1] (* _Harvey P. Dale, Mar 12 2013 *)
a[ n_] := SeriesCoefficient[ 1 / (1 - x) / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 09 2013 *)
Accumulate[PartitionsP[Range[0,49]]] (* George Beck, Oct 23 2014; typo fixed by Virgile Andreani, Jul 10 2016 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(m=1, n, 1 - x^m, 1 + x * O(x^n)) / (1 - x), n))}; /* Michael Somos, Nov 08 2002 */

x='x+O('x^66); Vec(1/((1-x)*eta(x))) /* Joerg Arndt, May 15 2011 */

a(n) = sum(k=0, n, numbpart(k)); \\ Michel Marcus, Sep 16 2016

from itertools import accumulate
def A000070iter(n):
    L = [0]*n; L[0] = 1
    def numpart(n):
        S = 0; J = n-1; k = 2
        while 0 <= J:
            T = L[J]
            S = S+T if (k//2)%2 else S-T
            J -= k  if (k)%2 else k//2
            k += 1
        return S
    for j in range(1, n): L[j] = numpart(j)
    return accumulate(L)
print(list(A000070iter(100))) # Peter Luschny, Aug 30 2019

# Using function A365676Row. Compare also A365675.
from itertools import accumulate
def A000070List(size: int) -> list[int]:
    return [sum(accumulate(reversed(A365676Row(n)))) for n in range(size)]
print(A000070List(45))  # Peter Luschny, Sep 16 2023

def A000070_list(leng):
    p = [number_of_partitions(n) for n in range(leng)]
    return [add(p[:k+1]) for k in range(leng)]
A000070_list(45) # Peter Luschny, Sep 15 2014

Euler transform of [ 2, 1, 1, 1, 1, 1, 1, ...].
log(a(n)) ~ -3.3959 + 2.44613*sqrt(n). - Robert G. Wilson v, Jan 11 2002
a(n) = (1/n)*Sum_{k=1..n} (sigma(k)+1)*a(n-k), n > 1, a(0) = 1. - Vladeta Jovovic, Aug 22 2002
G.f.: (1/(1 - x))*Product_{m >= 1} 1/(1 - x^m).
a(n) seems to have the same parity as A027349(n+1). Comment from James Sellers, Mar 08 2006: that is true.
a(n) = A000041(2n+1) - A110618(2n+1) = A000041(2n+2) - A110618(2n+2). - Henry Bottomley, Aug 01 2005
Row sums of triangle A133735. - Gary W. Adamson, Sep 22 2007
a(n) = A092269(n+1) - A195820(n+1). - Omar E. Pol, Oct 20 2011
a(n) = A181187(n+1,1) - A181187(n+1,2). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 23 2013: (Start)
Gupta gives the asymptotic result a(n-1) ~ sqrt(6/Pi^2)* sqrt(n)*p(n), where p(n) is the partition function A000041(n).
Let P(2,n) denote the set of partitions of n into parts k >= 2.
a(n-2) = Sum_{parts k in all partitions in P(2,n)} phi(k), where phi(k) is the Euler totient function (see A000010). Using this result and Mertens's theorem on the average order of the phi function, leads to the asymptotic result
a(n-2) ~ (6/Pi^2)*n*(p(n) - p(n-1)) = (6/Pi^2)*A138880(n) as n -> infinity. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)) * (1 + 11*Pi/(24*sqrt(6*n)) + (73*Pi^2 - 1584)/(6912*n)). - Vaclav Kotesovec, Oct 26 2016
a(n) = A024786(n+2) + A024786(n+1). - Vaclav Kotesovec, Nov 05 2016
G.f.: exp(Sum_{k>=1} (sigma_1(k) + 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
a(n) = A025065(2n). - Gus Wiseman, Oct 26 2018
a(n - 1) = A000041(2n) - A209816(n). - Gus Wiseman, Oct 26 2018

A066633 Triangle T(n,k), n >= 1, 1 <= k <= n, giving number of k's in all partitions of n.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 7, 3, 1, 1, 12, 4, 2, 1, 1, 19, 8, 4, 2, 1, 1, 30, 11, 6, 3, 2, 1, 1, 45, 19, 9, 6, 3, 2, 1, 1, 67, 26, 15, 8, 5, 3, 2, 1, 1, 97, 41, 21, 13, 8, 5, 3, 2, 1, 1, 139, 56, 31, 18, 12, 7, 5, 3, 2, 1, 1, 195, 83, 45, 28, 17, 12, 7, 5, 3, 2, 1, 1, 272, 112, 63, 38, 25, 16, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Naohiro Nomoto, Jan 09 2002

It appears that row n lists the first differences of the row n of triangle A181187 together with 1 (as the final term of the row n). - Omar E. Pol, Feb 26 2012
It appears that reversed rows converge to A000041. - Omar E. Pol, Mar 11 2012
Proof: For a partition of n with k>floor(n/2+1), k can only occur as the largest part; the other parts sum to n-k, so that T(n,n-k)=A000041(k). - George Beck, Jun 30 2019
T(n,k) is also the total number k's that are divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. - Omar E. Pol, Feb 05 2021

			For n = 3, k = 1; 3 = 2+1 = 1+1+1. T(3,1) = (zero 1's) + (one 1's) + (three 1's), so T(3,1) = 4.
Triangle begins:
    1;
    2,   1;
    4,   1,  1;
    7,   3,  1,  1;
   12,   4,  2,  1,  1;
   19,   8,  4,  2,  1,  1;
   30,  11,  6,  3,  2,  1,  1;
   45,  19,  9,  6,  3,  2,  1, 1;
   67,  26, 15,  8,  5,  3,  2, 1, 1;
   97,  41, 21, 13,  8,  5,  3, 2, 1, 1;
  139,  56, 31, 18, 12,  7,  5, 3, 2, 1, 1;
  195,  83, 45, 28, 17, 12,  7, 5, 3, 2, 1, 1;
  272, 112, 63, 38, 25, 16, 11, 7, 5, 3, 2, 1, 1;
  ...

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.2.1.5, Problem 73(b), pp. 415, 761. - N. J. A. Sloane, Dec 30 2018

Alois P. Heinz, Rows n = 1..141, flattened
Manosij Ghosh Dastidar and Sourav Sen Gupta, Generalization of a few results in Integer Partitions, arXiv preprint arXiv:1111.0094 [cs.DM], 2011.
Joseph Vandehey, Digital problems in the theory of partitions, Integers (2024) Vol. 24A, Art. No. A18. See p. 3.
Eric Weisstein's World of Mathematics, Elder's Theorem

Row sums give positive terms of A006128.

Columns (1-10): A000070, A024786-A024794.

b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1+n*x, b(n, i-1)+
      `if`(i>n, 0, (g->g+coeff(g, x, 0)*x^i)(b(n-i, i))))
    end:
T:= n-> (p->seq(coeff(p, x, i), i=1..n))(b(n, n)):
seq(T(n), n=1..14);  # Alois P. Heinz, Mar 21 2012

Table[Count[Flatten[IntegerPartitions[n]],k],
{n,1,20},{k,1,n}]
TableForm[% ] (* as a triangle *)
Flatten[%%]   (* as a sequence *)
(* Clark Kimberling, Mar 03 2010 *)
T[n_, n_] = 1; T[n_, k_] /; k, ] = 0; Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 21 2015, after Omar E. Pol *)

from math import isqrt, comb
from sympy import partition
def A066633(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    b = n-comb(a,2)
    return sum(partition(j) for j in range(a%b,a,b)) # Chai Wah Wu, Nov 13 2024

G.f. for the number of k's in all partitions of n is x^k/(1-x^k)* Product_{m>=1} 1/(1-x^m). - Vladeta Jovovic, Jan 15 2002

T(n, k) = Sum_{j

Equals triangle A027293 * A051731 as infinite lower triangular matrices. - Gary W. Adamson Mar 21 2011

It appears that T(n+k,k) = T(n,k) + A000041(n). - Omar E. Pol, Feb 04 2012. This was proved in the Dastidar-Gupta paper in Lemma 1. - George Beck, Jun 26 2019

It appears that T(n,k) = A206563(n,k) - A206563(n,k+2). - Omar E. Pol, Feb 26 2012

T(n,k) = Sum_{j=1..n} A182703(j,k). - Omar E. Pol, May 02 2012

More terms from Vladeta Jovovic, Jan 11 2002

Showing 1-10 of 25 results. Next

cp's OEIS Frontend

A145975 Triangle read by rows, partition triangle A027293 convolved with A010815.

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A146023 Triangle read by rows, square of A027293.

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A091114 Number of partitions of n-th composite number containing the smallest prime factor: a(n) = A027293(A002808(n), A056608(n)).

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PARI

A143866 Eigentriangle of A027293.

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A152538 Triangle read by rows, A027293 * (A152537 * 0^(n-k)).

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A174740 Triangle read by rows, A027293 * an infinite lower triangular matrix with A147843 as the diagonal and the rest zeros.

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A147768 Triangle read by rows: A000012^(-2) * A027293 as infinite lower triangular matrices.

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

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A000070 a(n) = Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).

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