A008967 -id:A008967 - cp's OEIS frontend

A047971 Triangle of coefficients of Gaussian polynomials [ n+3,3 ].

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 5, 6, 6, 6, 6, 5, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 5, 7, 7, 8, 8, 8, 7, 7, 5, 4, 3, 2, 1, 1

Offset: 0

N. J. A. Sloane

a(n) as illustrated is related to the following sequences: The row sum values are A001400. The column sums are A000292. The row lengths are the stuttering sequence A037915 (stutter values in A016777). The column lengths are the sequence A016777. The max values in each column are A001971. - Alford Arnold, Aug 16 2004
The Gaussian polynomial (or Gaussian binomial) [n,3]q is an example of a q-binomial coefficient (see the link) and may be defined for n >= 3 by [n,3]_q = ([n]_q * [n-1]_q * [n-2]_q)/([1]_q * [2]_q * [3]_q), where [n]_q := q^n - 1. The first few values are: [3,3]_q = 1; [4,3]_q = 1 + q + q^2 + q^3; [5,3]_q = 1 + q + 2q^2 + 2q^3 + 2q^4 + q^5 + q^6. - _Peter Bala, Sep 23 2007
The entry a(p,w), p >= 0, w = 0,1,...,3*p, of this irregular triangle is the number of nonnegative solutions of m_0 + m_1 + m_2 + m_3 = p and 1*m_1 + 2*m_2 + 3*m_3 = w. See the Hawkins reference given in A008967, p. 264, (4,7),(4.8), concerning Cayley's counting problem.  N(p,3,w) there equals a(p,w). The o.g.f. has been given in the formula section by Peter Bala.  See also the Cayley reference given in A008967, p. 110, 35. with m = 3, Theta = p and q = w. - Wolfdieter Lang, Dec 02 2012
The entry a(p,w) p >= 0, w = 0,1,...,3*p, of this array gives the number of partitions of w into at most p parts, each at most 3. This follows from the preceding comment with the two Diophantine equations. From Andrews, p. 33 and p. 35, a(p,w) (called there p(N,M,n) with N=p, M=3, n=w) gives also the number of partitions of w into at most 3 parts, each at most p. This is in accordance with the symmetry of the q-binomials [p+3,p] = [p+3,3]. - Wolfdieter Lang, Dec 04 2012

			The table a(p,w) = [q^w][p+3,3]_q starts:
p\w 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
0:  1
1:  1  1  1  1
2:  1  1  2  2  2  1  1
3:  1  1  2  3  3  3  3  2  1  1
4:  1  1  2  3  4  4  5  4  4  3  2  1  1
5:  1  1  2  3  4  5  6  6  6  6  5  4  3  2  1  1
6:  1  1  2  3  4  5  7  7  8  8  8  7  7  5  4  3  2  1  1
... Reformatted and extended by _Wolfdieter Lang_, Dec 04 2012
Partition example: Row p=2 is 1 1 2 2 2 1 1 because there are ten solution for (m_0, m_1, m_2, m_3) of the first equation given in a comment above, namely (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (1,1,0,0), (1,0,1,0), (1,0,0,1), (0,1,1,0), (0,1,0,1) and (0,0,1,1) which have the w = 1*m_1 + 2*m_2 + 3*m_3 values 0, 2, 4, 6, 1, 2, 3, 3, 4 and 5, respectively. Therefore there are 1, 1, 2, 2, 2, 1, 1 solutions for w = 0, 1, 2, 3, 4, 5, 6, respectively. - _Wolfdieter Lang_, Dec 03 2012
a(4,5) = 4 because there are 4 partitions of 5 with 1, 2, 3  or 4 parts, each being <= 3, namely all partitions of 5 excluding 5, 14 and 11111. There are also 4 partitions of 5 with 1, 2, or 3 parts, each being <= 4, namely all partitions of 5 excluding 5, 1112 and 11111. - _Wolfdieter Lang_, Dec 04 2012
The table may also be arranged as follows (see the Alford Arnold comment above):
1
..1
..1..1
..1..1..1
..1..2..1..1
.....2..2..1..1
.....2..3..2..1..1
.....1..3..3..2..1..1

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 242.

Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

Cf. A008967.

Cf. A001400.

nmax = 6;
se = Series[ 1/Product[1 - q^k*x, {k, 0, 3}], {x, 0, nmax}];
row[n_] := CoefficientList[ SeriesCoefficient[se, n], q];
Flatten[ Table[ row[n], {n, 0, nmax}]] (* Jean-François Alcover, Dec 19 2011 *)

O.g.f.: 1/((1-x)(1-qx)(1-q^2x)(1-q^3x)) = 1 + x(1 + q + q^2 + q^3) + x^2(1 + q + 2q^2 + 2q^3 + 2q^4 + q^5 + q^6) + .... - Peter Bala, Sep 23 2007

A365825 Number of integer partitions of n that are not of length 2 and do not contain n/2.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 6, 12, 14, 26, 31, 51, 61, 95, 114, 169, 202, 289, 347, 481, 576, 782, 936, 1244, 1487, 1946, 2323, 2997, 3570, 4551, 5414, 6827, 8103, 10127, 11997, 14866, 17575, 21619, 25507, 31166, 36692, 44563, 52362, 63240, 74152, 89112, 104281, 124731

Offset: 0

Gus Wiseman, Sep 19 2023

nonn

Also the number of integer partitions of n with no two possibly equal parts summing to n.

			The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)        (8)
            (111)  (1111)  (221)    (222)     (322)      (332)
                           (311)    (411)     (331)      (521)
                           (2111)   (2211)    (421)      (611)
                           (11111)  (21111)   (511)      (2222)
                                    (111111)  (2221)     (3221)
                                              (3211)     (3311)
                                              (4111)     (5111)
                                              (22111)    (22211)
                                              (31111)    (32111)
                                              (211111)   (221111)
                                              (1111111)  (311111)
                                                         (2111111)
                                                         (11111111)

First condition alone is A058984, complement A004526, ranks A100959.
Second condition alone is A086543, complement A035363, ranks !A344415.
The complement is counted by A238628.
The strict case is A365826, complement A365659.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A140106 counts strict partitions of length 2, complement A365827.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A005408, A008967, A068911, A365377, A365544, A365543.

Table[Length[Select[IntegerPartitions[n],Length[#]!=2&&FreeQ[#,n/2]&]],{n,0,15}]

from sympy import npartitions
def A365825(n): return npartitions(n)-(m:=n>>1)-(0 if n&1 else npartitions(m)-1) # Chai Wah Wu, Sep 23 2023

Heinz numbers are A100959 /\ !A344415.

a(n) = A000041(n)-(n-1)/2 if n is odd. a(n) = A000041(n)-n/2-A000041(n/2)+1 if n is even. - Chai Wah Wu, Sep 23 2023

a(31)-a(47) from Chai Wah Wu, Sep 23 2023

A365827 Number of strict integer partitions of n whose length is not 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 16, 20, 25, 30, 38, 45, 55, 66, 79, 93, 111, 130, 153, 179, 209, 242, 282, 325, 375, 432, 496, 568, 651, 742, 846, 963, 1094, 1240, 1406, 1589, 1795, 2026, 2282, 2567, 2887, 3240, 3634, 4072, 4557, 5094, 5692, 6351

Offset: 0

Gus Wiseman, Sep 20 2023

nonn

Also the number of strict integer partitions of n with no pair of distinct parts summing to n.

			The a(5) = 1 through a(13) = 12 strict partitions (A..D = 10..13):
  (5)  (6)    (7)    (8)    (9)    (A)     (B)     (C)     (D)
       (321)  (421)  (431)  (432)  (532)   (542)   (543)   (643)
                     (521)  (531)  (541)   (632)   (642)   (652)
                            (621)  (631)   (641)   (651)   (742)
                                   (721)   (731)   (732)   (751)
                                   (4321)  (821)   (741)   (832)
                                           (5321)  (831)   (841)
                                                   (921)   (931)
                                                   (5421)  (A21)
                                                   (6321)  (5431)
                                                           (6421)
                                                           (7321)

The complement is counted by A140106 shifted left.
Heinz numbers are A005117 \ A006881 = A005117 /\ A100959.
The non-strict version is A058984, complement A004526.
The case not containing n/2 is A365826, non-strict A365825.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A008967, A035363, A078408, A365659.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[#]!=2&]],{n,0,30}]

a(n) = A000009(n) - A004526(n-1) for n > 0.

A366131 Number of subsets of {1..n} with two elements (possibly the same) summing to n.

Original entry on oeis.org

0, 0, 2, 2, 10, 14, 46, 74, 202, 350, 862, 1562, 3610, 6734, 14926, 28394, 61162, 117950, 249022, 484922, 1009210, 1979054, 4076206, 8034314, 16422922, 32491550, 66045982, 131029082, 265246810, 527304974, 1064175886, 2118785834, 4266269482, 8503841150, 17093775742, 34101458042, 68461196410, 136664112494

Offset: 0

Gus Wiseman, Oct 07 2023

			The a(0) = 0 through a(5) = 14 subsets:
  .  .  {1}    {1,2}    {2}        {1,4}
        {1,2}  {1,2,3}  {1,2}      {2,3}
                        {1,3}      {1,2,3}
                        {2,3}      {1,2,4}
                        {2,4}      {1,3,4}
                        {1,2,3}    {1,4,5}
                        {1,2,4}    {2,3,4}
                        {1,3,4}    {2,3,5}
                        {2,3,4}    {1,2,3,4}
                        {1,2,3,4}  {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {2,3,4,5}
                                   {1,2,3,4,5}

Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

The complement is counted by A117855.
For pairs summing to n + 1 we have A167936.
A068911 counts subsets of {1..n} w/o two distinct elements summing to n.
A093971/A088809/A364534 count certain types of sum-full subsets.
Cf. A008967, A167762, A238628, A365376, A365377, A365381, A365541, A365544.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Tuples[#,2],n]&]],{n,0,10}]

def A366131(n): return (1<>1)<<1) if n else 0 # Chai Wah Wu, Nov 14 2023

From Chai Wah Wu, Nov 14 2023: (Start)
a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3) for n > 3.
G.f.: 2*x^2*(1 - x)/((2*x - 1)*(3*x^2 - 1)). (End)

A118175 Binary representation of n-th iteration of the Rule 220 elementary cellular automaton starting with a single black cell.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0

Offset: 0

Eric W. Weisstein, Apr 13 2006

From Franklin T. Adams-Watters, Jul 05 2009: (Start)
Divided into rows of length 2n, row n consists of n 1's followed by n 0's.
Characteristic function of A061885, 1-based characteristic function of A004201. (End)
From Wolfdieter Lang, Dec 05 2012: (Start)
The row lengths sequence is A005408 (the odd numbers). The sum of row No. n is given by A000027(n+1).
This table is the first difference table of the q-binomial (Gauss polynomial) coefficient table G(2;n,k) = [q^k]( [n+2,2]_q) (see table A008967): a(n,k) = G(2;n,k) - G(2;n-1,k). The o.g.f. for the row polynomials is therefore G2(q,z) = (1-z)/Product((1-q^j*z),j=0..2) = 1/((1-q*z)*(1-q^2*z)). Therefore, a(n,k) determines the number of partitions of k into precisely n parts, each <= 2. It determines also the number of partitions of k into at most 2 parts, each <= n but not <= (n-1), i.e., with part n present. See comments on A008967 regarding partitions.
From the o.g.f. G2(q,z) it should be clear that there are 0's for n > k and only 1's for k = n,...,2*n.
(End)
This sequence is also generated by Rule 252. - Robert Price, Jan 31 2016
a(n) is 1 if the nearest square to n is >= n, otherwise 0. - Branko Curgus, Apr 25 2017

			The table a(n,k) begins:
  n\k 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
  0:  1
  1:  0  1  1
  2:  0  0  1  1  1
  3:  0  0  0  1  1  1  1
  4:  0  0  0  0  1  1  1  1  1
  5:  0  0  0  0  0  1  1  1  1  1  1
  6:  0  0  0  0  0  0  1  1  1  1  1  1  1
  7:  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1
  8:  0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1  1
  9:  0  0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1  1  1
... Reformatted and extended by _Wolfdieter Lang_, Dec 05 2012
Partition examples: a(n,k) = 0 if n>k because the maximal number of parts of a partition of k is k. a(n,n) = 1, n >= 1, because only the partition 1^n has n parts, and 1 <= 2.
  a(2,3) = 1 because the only partition of 3 with 2 parts, each <= 2, is 1,2. Also, the only partition of 3 with at most 2 parts, each <= 2, and a part 2 present is also 1,2.
  a(5,7) =1 because the only 5-part partition of 7 with maximal part 2 is 1^3,2^3. Also, the only partition of 7 with at most 2 parts, each <= 5, which a part 5 present is 2,5.

Robert Price, Table of n, a(n) for n = 0..9999
Eric Weisstein's World of Mathematics, Rule 220
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Stephen Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A083420, A219238.

Table[1 - Ceiling[Sqrt[n]] + Round[Sqrt[n]], {n, 1, 257}] (* Branko Curgus, Apr 26 2017 *)
Table[{Array[1&,n],Array[0&,n]},{n,1,5}]//Flatten (* Wolfgang Hintze, Jul 28 2017 *)

from math import isqrt
def A118175(n): return 1+int(n-(m:=isqrt(n+1))*(m+1)>=0)-int(m**2!=n+1) # Chai Wah Wu, Jul 30 2022

a(n) = 1 - A079813(n+1). - Philippe Deléham, Jan 02 2012
a(n) = 1 - ceiling(sqrt(n+1)) + round(sqrt(n+1)). - Branko Curgus, Apr 27 2017 [Corrected by Ridouane Oudra, Dec 01 2019]
G.f.: x/(1 - x)*( Sum_{n >= 1} x^(n^2-n)*(1-x^n)) = 1/(2-2*x)* ( x + x^(3/4)*EllipticTheta(2,0,x) - x*EllipticTheta(3,0,x) ). - Wolfgang Hintze, Jul 28 2017
a(n) = floor(sqrt(n+1)+1/2) - floor(sqrt(n)) = round(sqrt(n+1)) - floor(sqrt(n)). - Ridouane Oudra, Dec 01 2019

A219237 Coefficient of Gauss polynomials [n+4,4]_q (q-binomials).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 1, 1, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1, 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 6, 8, 9, 11, 11, 12, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 6, 9, 10, 13, 14, 16, 16, 18, 16, 16, 14, 13, 10, 9, 6, 5, 3, 2, 1, 1

Offset: 0

Wolfdieter Lang, Dec 04 2012

The length of row n of this table is 4*n + 1 = A016813(n).
The sum of row n is binomial(n+4,4) = A000332(n+4), n>= 0.
The Gauss polynomial [n+4,4]q := [n+4]_q/([n]_q*[4]_q), with [n]_q = Product{j=1..n} (1-q^j) = (q;q)n (in q-shifted factorials notation), n>=0. [n+4,4]_q = (Product{j=(n+1)..(n+4)} (1-q^j))/(Product_{j=1..4} (1-q^j)). This is a polynomial in q (of degree 4*n) because it is the o.g.f. of the numbers p(n,4,k), the number of partitions of k into at most 4 parts, each <= n (see Andrews, p. 33 and 35). p(n,4,k) is also the number of partitions of k into at most n parts, each <= 4, due to the symmetry property [n+4,4]q = [n+4,n]_q (Andrews, (3,3,2), p.35). With the latter interpretation p(n,4,k) is the number of solutions of the two Diophantine equations Sum{j=1..4} j*m(j) = k and Sum_{j=0..m} m(j) = n, i.e. Sum_{j=1..m} m(j) = n - m(0), with 0 <= m(j) <= n. Therefore p(n,4,k) = [q^k] [x^n] G(4;x,q) with o.g.f. G(4;x,q) = 1/Product_{j=0..4} (1-x*q^j). Here we will call p(n,4,k) = T(n,k), n >= 0, 0 <= k <= 4*n.
See the comments in A008967 concerning a counting problem of Cayley (there m = 4, Theta = n and q = k), described also in the Hawkins reference (N(p->n,4,w->k) = T(n,k)) given there.

			The triangle T(n,k) begins:
  n\k 0  1  2  3  4  5  6   7   8   9  10  11  12  13  14  15  16 ...
  0:  1
  1:  1  1  1  1  1
  2:  1  1  2  2  3  2  2   1   1
  3:  1  1  2  3  4  4  5   4   4   3   2   1   1
  4:  1  1  2  3  5  5  7   7   8   7   7   5   5   3   2   1   1
  5:  1  1  2  3  5  6  8   9  11  11  12  11  11   9   8   6   5   3  2  1  1
  6:  1  1  2  3  5  6  9  10  13  14  16  16  18  16  16  14  13  10  9  6  5  3  2  1 1
Partition interpretation: T(3,5) = 4 because there are 4 partitions of 5 into at most 4 parts, each <= 3, namely 23, 113, 122 and 1112. here are also 4 partitions of 5 into at most 3 parts, each <= 4, namely 14, 23, 113 and 122. Note the conjugacy of the partitions 1112 and 14.
The 4 solutions of the two Diophantine equations given in a comment, with k=5 and n=3, are for (m(0), m(1), m(2), m(3), m(4)): (1,1,0,0,1), (1,0,1,1,0), (0,2,0,1,0) and (0,1,2,0,0).

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 240, 242-3.

Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

Cf. A000012 (as triangle for m=1), A008967 (m=2), A047971 (m=3).

Cf. A000332, A016813.

a[0, 0] = 1; a[n_, k_] := SeriesCoefficient[ QBinomial[n+4, 4, q], {q, 0, k}]; Table[a[n, k], {n, 0, 6}, {k, 0, 4*n}] // Flatten (* Jean-François Alcover, Dec 04 2013 *)

T(n,k) = [q^k] [x^n](1/Product_{j=0..4} (1-x*q^j)), n >= 0, 0 <= k <= 4*n.
T(n,k) = [q^k]([n+4,4]_q), n >= 0, 0 <= k <= 4*n.
See the comments above.

A366130 Number of subsets of {1..n} with a subset summing to n + 1.

Original entry on oeis.org

0, 0, 1, 2, 7, 15, 38, 79, 184, 378, 823, 1682, 3552, 7208, 14948, 30154, 61698, 124302, 252125, 506521, 1022768, 2051555, 4127633, 8272147, 16607469, 33258510, 66680774, 133467385, 267349211, 535007304, 1071020315, 2142778192, 4288207796

Offset: 0

Gus Wiseman, Oct 07 2023

			The subset S = {1,2,4} has subset {1,4} with sum 4+1 and {2,4} with sum 5+1 and {1,2,4} with sum 6+1, so S is counted under a(4), a(5), and a(6).
The a(0) = 0 through a(5) = 15 subsets:
  .  .  {1,2}  {1,3}    {1,4}      {1,5}
               {1,2,3}  {2,3}      {2,4}
                        {1,2,3}    {1,2,3}
                        {1,2,4}    {1,2,4}
                        {1,3,4}    {1,2,5}
                        {2,3,4}    {1,3,5}
                        {1,2,3,4}  {1,4,5}
                                   {2,3,4}
                                   {2,4,5}
                                   {1,2,3,4}
                                   {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {2,3,4,5}
                                   {1,2,3,4,5}

For pairs summing to n + 1 we have A167762, complement A038754.
For n instead of n + 1 we have A365376, for pairs summing to n A365544.
The complement is counted by A365377 shifted.
The complement for pairs summing to n is counted by A365377.
A068911 counts subsets of {1..n} w/o two distinct elements summing to n.
A093971/A088809/A364534 count certain types of sum-full subsets.
Cf. A004526, A004737, A008967, A046663, A238628, A365381, A365541.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#],n+1]&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A366130(n):
    a = tuple(set(p.keys()) for p in partitions(n+1,k=n) if max(p.values(),default=0)==1)
    return sum(1 for k in range(2,n+1) for w in (set(d) for d in combinations(range(1,n+1),k)) if any(s<=w for s in a)) # Chai Wah Wu, Nov 24 2023

Diagonal k = n + 1 of A365381.

a(20)-a(32) from Chai Wah Wu, Nov 24 2023

A106822 Triangle read by rows: g.f. for row r is Product_{i=1..r-2} (x^i-x^(r+1))/(1-x^i).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, May 20 2005

			Initial rows are:
[1]
[1]
[0, 1, 1, 1]
[0, 0, 0, 1, 1, 2, 1, 1]
[0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 1, 1]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 2, 2, 1, 1]

See A008967 for references.

G. C. Greubel, Rows n = 0..25 of the irregular triangle, flattened

If the initial zeros in each row are omitted, we get A008967.

Cf. A008967, A106823.

f2:=r->mul( (x^i-x^(r+1))/(1-x^i), i = 1..r-2); for r from 1 to 10 do series(f2(r),x,50); od:

f[n_, x_]:= Product[(x^j - x^(n+2))/(1 - x^j), {j, n-1}];
T[n_]:= CoefficientList[f[n, x], x];
Table[T[n], {n, 0, 10}]//Flatten (* G. C. Greubel, Sep 12 2021 *)

row(r) = Vecrev(prod(i=1, r-2, (x^i-x^(r+1))/(1-x^i))); \\ Michel Marcus, Sep 14 2021

A365660 Number of integer partitions of 2n with exactly n distinct sums of nonempty submultisets.

Original entry on oeis.org

1, 1, 1, 3, 2, 6, 6, 16, 12, 20, 26, 59, 45, 79, 94, 186, 142, 231, 244, 442, 470, 616, 746, 1340, 1053, 1548, 1852, 2780, 2826, 3874, 4320, 6617, 6286, 7924, 9178, 13180, 13634, 17494, 20356, 28220, 29176, 37188, 41932, 56037

Offset: 0

Gus Wiseman, Sep 16 2023

Are n = 1, 2, 4 the only n such that none of these partitions has 1?

Are n = 2, 4, 5, 8, 9 the only n such that none of these partitions is strict?

			The partition (433) has sums 3, 4, 6, 7, 10 so is counted under a(5).
The a(1) = 1 through a(7) = 16 partitions:
(2)  (2,2)  (4,2)    (4,2,2)    (4,3,3)      (6,4,2)        (6,5,3)
            (5,1)    (2,2,2,2)  (4,4,2)      (6,5,1)        (8,4,2)
            (2,2,2)             (6,2,2)      (4,4,2,2)      (8,5,1)
                                (8,1,1)      (6,2,2,2)      (9,3,2)
                                (4,2,2,2)    (4,2,2,2,2)    (9,4,1)
                                (2,2,2,2,2)  (2,2,2,2,2,2)  (10,3,1)
                                                            (11,2,1)
                                                            (4,4,4,2)
                                                            (5,3,3,3)
                                                            (6,4,2,2)
                                                            (8,2,2,2)
                                                            (11,1,1,1)
                                                            (4,4,2,2,2)
                                                            (6,2,2,2,2)
                                                            (4,2,2,2,2,2)
                                                            (2,2,2,2,2,2,2)

For n instead of 2n we have A126796.
Central column n = 2k of A365658.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A002219 counts partitions of 2n with a submultiset summing to n.
A046663 counts partitions of n w/o a submultiset of sum k, strict A365663.
A122768 counts distinct nonempty submultisets of partitions.
A299701 counts sums of submultisets of prime indices, of partitions A304792.
A364272 counts sum-full strict partitions, sum-free A364349.
A365543 counts partitions of n w/ a submultiset of sum k, strict A365661.
Cf. A000041, A008967, A095944, A364911, A365376, A365377.

msubs[y_]:=primeMS/@Divisors[Times@@Prime/@y];
Table[Length[Select[IntegerPartitions[2n], Length[Union[Total/@Rest[msubs[#]]]]==n&]],{n,0,10}]

from collections import Counter
from sympy.utilities.iterables import partitions, multiset_combinations
def A365660(n):
    c = 0
    for p in partitions(n<<1):
        q, s = list(Counter(p).elements()), set()
        for l in range(1,len(q)+1):
            for k in multiset_combinations(q,l):
                s.add(sum(k))
                if len(s) > n:
                    break
            else:
                continue
            break
        if len(s)==n:
            c += 1
    return c # Chai Wah Wu, Sep 20 2023

a(21)-a(38) from Chai Wah Wu, Sep 20 2023

a(39)-a(43) from Chai Wah Wu, Sep 21 2023

A365826 Number of strict integer partitions of n that are not of length 2 and do not contain n/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 20, 20, 30, 31, 45, 46, 66, 68, 93, 97, 130, 136, 179, 188, 242, 256, 325, 344, 432, 459, 568, 606, 742, 793, 963, 1031, 1240, 1331, 1589, 1707, 2026, 2179, 2567, 2766, 3240, 3493, 4072, 4393, 5094, 5501, 6351

Offset: 0

Gus Wiseman, Sep 20 2023

nonn

Also the number of strict integer partitions of n without two parts (allowing parts to be re-used) summing to n.

			The a(6) = 1 through a(12) = 7 strict partitions:
  (6)  (7)      (8)      (9)      (10)       (11)       (12)
       (4,2,1)  (5,2,1)  (4,3,2)  (6,3,1)    (5,4,2)    (5,4,3)
                         (5,3,1)  (7,2,1)    (6,3,2)    (7,3,2)
                         (6,2,1)  (4,3,2,1)  (6,4,1)    (7,4,1)
                                             (7,3,1)    (8,3,1)
                                             (8,2,1)    (9,2,1)
                                             (5,3,2,1)  (5,4,2,1)

The second condition alone has bisections A078408 and A365828.
The complement is counted by A365659.
The non-strict version is A365825, complement A238628.
The first condition alone is A365827, complement A140106.
A000041 counts integer partitions, strict A000009.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A004526, A005408, A008967, A035363, A058984, A086543, A100959, A344415, A365376, A365377, A365543.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Tuples[#,2],n]&]], {n,0,30}]

cp's OEIS Frontend

A047971 Triangle of coefficients of Gaussian polynomials [ n+3,3 ].

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A365825 Number of integer partitions of n that are not of length 2 and do not contain n/2.

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A365827 Number of strict integer partitions of n whose length is not 2.

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A366131 Number of subsets of {1..n} with two elements (possibly the same) summing to n.

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A118175 Binary representation of n-th iteration of the Rule 220 elementary cellular automaton starting with a single black cell.

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A219237 Coefficient of Gauss polynomials [n+4,4]_q (q-binomials).

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A366130 Number of subsets of {1..n} with a subset summing to n + 1.

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