A008967 -id:A008967 - cp's OEIS frontend

A106823 Triangle read by rows: g.f. for row r is Product( (x^i-x^(r+1))/(1-x^i), i = 1..r-2).

1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 2, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 6, 6, 6, 6, 5, 4, 3, 2, 1, 1

Offset: 0

N. J. A. Sloane, May 20 2005

			Initial rows are:
[1]
[1]
[1]
[0, 1, 1, 1, 1]
[0, 0, 0, 1, 1, 2, 2, 2, 1, 1]
[0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1]

See A008968 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 A008968.

Cf. A008967, A008968, A106822.

f3:=r->mul( (x^i-x^(r+1))/(1-x^i), i = 1..r-3); for r from 1 to 10 do series(f3(r),x,50); od:

f[n_, x_]:= Product[(x^j -x^(n+2))/(1-x^j), {j, n-2}];
T[n_]:= CoefficientList[f[n, x], x];
Table[T[n], {n, 0, 10}]//Flatten (* G. C. Greubel, Sep 14 2021 *)

A219238 Coefficient table for the first differences of table A047971: Coefficients of the difference of Gauss polynomials [n+3,3]_q - [n+2,3]_q.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 2, 2, 2, 1, 1, 0, 0, 0, 0, 1, 1, 2, 2, 3, 2, 2, 1, 1, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 4, 3, 3, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 4, 4, 3, 3, 2, 2, 1, 1

Offset: 0

Wolfdieter Lang, Dec 06 2012

The row lengths sequence is A016777 (3*n+1). The sum for row n is A000217(n+1) = binomial(n+2,2).
The coefficients of the Gauss polynomial [n+3,3]_q are given in A047971.
a(n,k) = [q^k]([n+3,3]_q - [n+2,3]_q). One can use the identity [n+3,3]_q - [n+2,3]_q = q^n*[n+2,2]_q (see the Andrews reference given in A047971, p. 35, (3.3.3)). Therefore, the present array is obtained from A008967 after a shift of row n by n units to the right, inserting zeros for the first n entries.
The o.g.f. of the row polynomials in q of degree 3*n is 1/((1-q)*(1-q^2)*(1-q^3)) (multiply the o.g.f. of A047971 by (1-z)). a(n,k) determines therefore the number of partitions of k with precisely n parts, each <= 3. Alternatively, a(n,k) determines the number of partitions of k with at most 3 parts, with each part <= n but not each part <= (n-1), i.e., part n, maybe more than once, is present besides possibly smaller ones.

			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  1
2:  0  0  1  1  2  1  1
3:  0  0  0  1  1  2  2  2  1  1
4:  0  0  0  0  1  1  2  2  3  2  2  1  1
5:  0  0  0  0  0  1  1  2  2  3  3  3  2  2  1  1
6:  0  0  0  0  0  0  1  1  2  2  3  3  4  3  3  2  2  1  1
...
Row n=1 is 0,1,1,1 because [3,2]_q = 1 + q + q^2 and the coefficient of q^{-1} is 0, the one of q^0 is 1, the one of q^1 is 1 and the one of q^2 is 1. A shift of row n=1 of A008967 by one unit to the right.
a(n,k) = 0 if n > k because a partition of k never has more than k parts.
a(n,k) = 0 if k > 3*n because there is no partition of 3*n+m, with m >= 1, and exactly n parts, each <= 3.
a(2,4) = 2 because the partitions of 4 with 2 parts are 1,3 and 2,2, and the parts in both are <= 3.
a(2,4) = 2 because the partitions of 4 with number of parts <= 3, each <= 2, are 2,2 and 1,1,2, and part 2 is present in both of them. Note the conjugacy of partitions 1,3 and 1,1,2.

Cf. A047971, A008967 (with shifted rows).

a(n,k) = [q^k]([n+3,3]_q - [n+2,3]_q), = [q^(k-n)] [n+2,2]_q , n >= 0, 0 <= k <= 3*n. For the Gauss polynomial (q-binomial) [n+m,m]_q = [m+n,n]_q see a comment on A219237 where also the Andrews reference and a link to Mathworld is found.

A089789 Number of irreducible factors of Gauss polynomials.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 0, 1, 2, 2, 1, 0, 0, 3, 3, 4, 3, 3, 0, 0, 1, 3, 3, 3, 3, 1, 0, 0, 3, 3, 5, 4, 5, 3, 3, 0, 0, 2, 4, 4, 5, 5, 4, 4, 2, 0, 0, 3, 4, 6, 5, 7, 5, 6, 4, 3, 0, 0, 1, 3, 4, 5, 5, 5, 5, 4, 3, 1, 0, 0, 5, 5, 7, 7, 9, 7, 9, 7, 7, 5, 5, 0, 0, 1, 5, 5, 6, 7, 7, 7, 7, 6, 5, 5, 1, 0

Offset: 0

Paul Boddington, Jan 09 2004

T(n,k) is the number of irreducible factors of the (separable) polynomial [n]!/([k]![n-k]!). Here [n]! denotes the product of the first n quantum integers, the n-th quantum integer being defined as (1-q^n)/(1-q).

T(n,k) gives the number of positive integers m <= n such that (n mod m) < (k mod m). - Tom Edgar, Aug 21 2014

			The triangle T(n,k) begins:
n\k  0  1  2  3  4  5  6  7  8  9  10  11  12  13 ...
0:   0
1:   0  0
2:   0  1  0
3:   0  1  1  0
4:   0  2  2  2  0
5:   0  1  2  2  1  0
6:   0  3  3  4  3  3  0
7:   0  1  3  3  3  3  1  0
8:   0  3  3  5  4  5  3  3  0
9:   0  2  4  4  5  5  4  4  2  0
10:  0  3  4  6  5  7  5  6  4  3   0
11:  0  1  3  4  5  5  5  5  4  3   1   0
12:  0  5  5  7  7  9  7  9  7  7   5   5   0
13:  0  1  5  5  6  7  7  7  7  6   5   5   1   0
... Formatted by _Wolfdieter Lang_, Dec 07 2012
T(8,3) equals the number of irreducible factors of (1-q^8)(1-q^7)(1-q^6)/((1-q^3)(1-q^2)(1-q)), which is a product of 5 cyclotomic polynomials in q, namely the 2nd, 4th, 6th, 7th and 8th. Thus T(8,3)=5.

Stan Wagon and Herbert S. Wilf, When are subset sums equidistributed modulo m?, The Electronic Journal of Combinatorics, Vol. 1, 1994 (#R3).

Cf. A008967, A047971, A219237, A000005.

T(n, k) = T(n-1, k-1) + d(n) - d(k), where d(n) is the number of divisors of n.

A365829 Squarefree non-semiprimes.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, 53, 59, 61, 66, 67, 70, 71, 73, 78, 79, 83, 89, 97, 101, 102, 103, 105, 107, 109, 110, 113, 114, 127, 130, 131, 137, 138, 139, 149, 151, 154, 157, 163, 165, 167, 170, 173, 174, 179, 181, 182, 186

Offset: 1

Gus Wiseman, Oct 05 2023

nonn

First differs from A030059 in having 210.

			The terms together with their prime indices begin:
     1: {}          43: {14}       102: {1,2,7}
     2: {1}         47: {15}       103: {27}
     3: {2}         53: {16}       105: {2,3,4}
     5: {3}         59: {17}       107: {28}
     7: {4}         61: {18}       109: {29}
    11: {5}         66: {1,2,5}    110: {1,3,5}
    13: {6}         67: {19}       113: {30}
    17: {7}         70: {1,3,4}    114: {1,2,8}
    19: {8}         71: {20}       127: {31}
    23: {9}         73: {21}       130: {1,3,6}
    29: {10}        78: {1,2,6}    131: {32}
    30: {1,2,3}     79: {22}       137: {33}
    31: {11}        83: {23}       138: {1,2,9}
    37: {12}        89: {24}       139: {34}
    41: {13}        97: {25}       149: {35}
    42: {1,2,4}    101: {26}       151: {36}

First condition alone is A005117 (squarefree).
Second condition alone is A100959 (non-semiprime).
The nonprime case is 1 followed by A350352.
Partitions of this type are counted by A365827, non-strict A058984.
A001358 lists semiprimes, squarefree A006881.
Cf. A000009, A004526, A008967, A078408, A365659.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]!=2&]

isok(k) = my(f=factor(k)); issquarefree(f) && (bigomega(f) != 2); \\ Michel Marcus, Oct 07 2023

Intersection of A005117 and A100959.

Complement of A001358 in A005117.

A045430 Number of dominoes with n spots (in standard set).

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Aug 23 2000

			a(3)=2 since there are two dominoes with three spots: 2,1 and 3,0.

Index entries for sequences related to dominoes

Cf. A008967 where rows are equivalent sequences for zero to n spots in each half-domino (in this standard case: 6)

A384157 Irregular triangle read by rows: T(n,k) is the number of connected induced k-vertex subgraphs of the hyperoctahedral graph of dimension n >= 1 up to automorphisms of the hyperoctahedral graph; 0 <= k <= 2*n.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 3, 2, 2, 1, 1, 1, 1, 1, 2, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 2, 3, 3, 4, 4, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 2, 3, 3, 4, 4, 5, 4, 4, 3, 3, 2, 2, 1, 1

Offset: 1

Peter Kagey and Pontus von Brömssen, May 21 2025

In this sequence, the empty graph is considered to be connected.
There are n!*2^n graph automorphisms of the n-hyperoctahedral graph.
The hyperoctahedral graph is also called the "cocktail party graph," and corresponds to the 1-skeleton of the n-dimensional cross-polytope.
Row 3 corresponds to the number of polyominoes on the faces of a cube up to rotation and reflection of the cube.
More generally, this sequence gives the number of k-celled polyforms whose cells are (n-1)-dimensional facets of the n-dimensional hypercube.
An induced subgraph of the hyperoctahedral graph is completely determined (up to automorphisms of the hyperoctahedral graph) by the number i of pairs of antipodal vertices and the number j of vertices whose antipode is not in the subgraph. The subgraph is disconnected if and only if i=1 and j=0. This implies a close relation to A008967 (which also counts disconnected subgraphs); see formula.

			Triangle begins:
  1 | 1, 1, 0;
  2 | 1, 1, 1, 1, 1;
  3 | 1, 1, 1, 2, 2, 1, 1;
  4 | 1, 1, 1, 2, 3, 2, 2, 1, 1;
  5 | 1, 1, 1, 2, 3, 3, 3, 2, 2, 1, 1;
  6 | 1, 1, 1, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1;
  7 | 1, 1, 1, 2, 3, 3, 4, 4, 4, 3, 3, 2, 2, 1, 1;
  8 | 1, 1, 1, 2, 3, 3, 4, 4, 5, 4, 4, 3, 3, 2, 2, 1, 1;
  ...

Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Wikipedia, Cross-polytope

Cf. A008967 (includes disconnected subgraphs), A369605 (hypercube graph), A383973 (edges).

T(n,k) = A008967(n+4,k) if k != 2; T(n,2) = A008967(n+4,2)-1.

G.f.: 1/((1-y)*(1-x*y)*(1-x^2*y)) - x^2*y/(1-y) - 1.

cp's OEIS Frontend

A106823 Triangle read by rows: g.f. for row r is Product( (x^i-x^(r+1))/(1-x^i), i = 1..r-2).

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A219238 Coefficient table for the first differences of table A047971: Coefficients of the difference of Gauss polynomials [n+3,3]_q - [n+2,3]_q.

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A089789 Number of irreducible factors of Gauss polynomials.

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A365829 Squarefree non-semiprimes.

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A045430 Number of dominoes with n spots (in standard set).

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A384157 Irregular triangle read by rows: T(n,k) is the number of connected induced k-vertex subgraphs of the hyperoctahedral graph of dimension n >= 1 up to automorphisms of the hyperoctahedral graph; 0 <= k <= 2*n.

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