A013609 -id:A013609 - cp's OEIS frontend

A317501 Triangle read by rows: T(0,0) = 1; T(n,k) = 2*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.

1, 2, 4, 8, 16, 1, 32, 4, 64, 12, 128, 32, 256, 80, 1, 512, 192, 6, 1024, 448, 24, 2048, 1024, 80, 4096, 2304, 240, 1, 8192, 5120, 672, 8, 16384, 11264, 1792, 40, 32768, 24576, 4608, 160, 65536, 53248, 11520, 560, 1, 131072, 114688, 28160, 1792, 10, 262144, 245760, 67584, 5376, 60

Offset: 0

Zagros Lalo, Sep 03 2018

Unsigned version of the triangle in A317506.
The numbers in rows of the triangle are along a "third layer" skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and along a "third layer" skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)
The coefficients in the expansion of 1/(1-2*x-x^4) are given by the sequence generated by the row sums.
The row sums give A008999.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.106919340376..., when n approaches infinity.

			Triangle begins:
       1;
       2;
       4;
       8;
      16,      1;
      32,      4;
      64,     12;
     128,     32;
     256,     80,     1;
     512,    192,     6;
    1024,    448,    24;
    2048,   1024,    80;
    4096,   2304,   240,    1;
    8192,   5120,   672,    8;
   16384,  11264,  1792,   40;
   32768,  24576,  4608,  160;
   65536,  53248, 11520,  560,  1;
  131072, 114688, 28160, 1792, 10;
  262144, 245760, 67584, 5376, 60;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Row sums give A008999.

Cf. A013609, A038207, A128099, A207538, A317506.

t[n_, k_] := t[n, k] = 2^(n - 4 k)/((n - 4 k)! k!) (n - 3 k)!; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/4]} ] // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 4, k - 1]]; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/4]}] // Flatten

T(n,k) = 2^(n - 4*k) / ((n - 4*k)! k!) * (n - 3*k)! where n >= 0 and 0 <= k <= floor(n/4).

A350749 Triangle read by rows: T(n,k) is the number of oriented graphs on n labeled nodes with k arcs, n >= 0, k = 0..n*(n-1)/2.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 12, 8, 1, 12, 60, 160, 240, 192, 64, 1, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024, 1, 30, 420, 3640, 21840, 96096, 320320, 823680, 1647360, 2562560, 3075072, 2795520, 1863680, 860160, 245760, 32768

Offset: 0

Andrew Howroyd, Feb 15 2022

			Triangle begins:
  [0] 1;
  [1] 1;
  [2] 1,  2;
  [3] 1,  6,  12,   8;
  [4] 1, 12,  60, 160,  240,  192,    64;
  [5] 1, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1350 (rows 0..20)

Row sums are A047656.
The unlabeled version is A350733.
Cf. A013609, A350732 (weakly connected), A350731 (strongly connected).

PARI
```
T(n,k) = 2^k * binomial(n*(n-1)/2, k)
```

row(n) = {Vecrev((1+2*y)^(n*(n-1)/2))}
{ for(n=0, 6, print(row(n))) }

T(n,k) = 2^k * binomial(n*(n-1)/2, k) = A013609(n*(n-1)/2, k).

T(n,k) = [y^k] (1+2*y)^(n*(n-1)/2).

A191897 Coefficients of the Z(n,x) polynomials; Z(0,x) = 1, Z(1,x) = x and Z(n,x) = xZ(n-1,x) - 2Z(n-2,x), n >= 2.

Original entry on oeis.org

1, 1, 0, 1, 0, -2, 1, 0, -4, 0, 1, 0, -6, 0, 4, 1, 0, -8, 0, 12, 0, 1, 0, -10, 0, 24, 0, -8, 1, 0, -12, 0, 40, 0, -32, 0, 1, 0, -14, 0, 60, 0, -80, 0, 16, 1, 0, -16, 0, 84, 0, -160, 0, 80, 0, 1, 0, -18, 0, 112, 0, -280, 0, 240, 0, -32

Offset: 0

Paul Curtz, Jun 19 2011

The coefficients of the Z(n,x) polynomials by decreasing exponents, see the formulas, define this triangle.

			The first few rows of the coefficients of the Z(n,x) are
  1;
  1,    0;
  1,    0,   -2;
  1,    0,   -4,    0;
  1,    0,   -6,    0,    4;
  1,    0,   -8,    0,   12,    0;
  1,    0,  -10,    0,   24,    0,   -8;
  1,    0,  -12,    0,   40,    0,  -32,    0;
  1,    0,  -14,    0,   60,    0,  -80,    0,   16;
  1,    0,  -16,    0,   84,    0, -160,    0,   80,    0;

Row sums: A107920(n+1). Main diagonal: A077966(n).
Z(n,x=1) = A107920(n+1), Z(n,x=2)  = A009545(n+1),
Z(n,x=3) = A000225(n+1), Z(n,x=4)  = A007070(n),
Z(n,x=5) = A107839(n),   Z(n,x=6)  = A154244(n),
Z(n,x=7) = A186446(n),   Z(n,x=8)  = A190975(n+1),
Z(n,x=9) = A190979(n+1), Z(n,x=10) = A190869(n+1).
Row sum without sign: A113405(n+1).
Cf. A128099, A013609.

nmax:=10: Z(0, x):=1 : Z(1, x):=x: for n from 2 to nmax do Z(n, x) := x*Z(n-1, x) - 2*Z(n-2, x) od: for n from 0 to nmax do for k from 0 to n do T(n, k) := coeff(Z(n, x), x, n-k) od: od: seq(seq(T(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jun 27 2011, revised Nov 29 2012

a[n_, k_] := If[OddQ[k], 0, 2^(k/2)*Coefficient[ ChebyshevU[n, x/2], x, n-k]]; Flatten[ Table[ a[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-François Alcover, Aug 02 2012, from 2nd formula *)

Z(0,x) = 1, Z(1,x) = x and Z(n,x) = x*Z(n-1,x) - 2*Z(n-2,x), n >= 2.

a(n,k) = A077957(k) * A053119(n,k). - Paul Curtz, Sep 30 2011

Edited and information added by Johannes W. Meijer, Jun 27 2011

A265014 Triangle read by rows: T(n,k) = number of neighbors in n-dimensional lattice for generalized neighborhood given with parameter k.

Original entry on oeis.org

2, 4, 8, 6, 18, 26, 8, 32, 64, 80, 10, 50, 130, 210, 242, 12, 72, 232, 472, 664, 728, 14, 98, 378, 938, 1610, 2058, 2186, 16, 128, 576, 1696, 3488, 5280, 6304, 6560, 18, 162, 834, 2850, 6882, 12258, 16866, 19170, 19682, 20, 200, 1160, 4520, 12584, 26024, 41384, 52904, 58024, 59048

Offset: 1

Dmitry Zaitsev, Nov 30 2015

In an n-dimensional hypercube lattice, the sequence gives the number of nodes situated at a Chebyshev distance of 1 combined with Manhattan distance not greater than k, 1<=k<=n. In terms of cellular automata, it gives the number of neighbors in a generalized neighborhood given with parameter k: at k=1, we obtain von Neumann's neighborhood with 2n neighbors (A005843), and at k=n, we obtain Moore's neighborhood with 3^n-1 neighbors (A024023). It represents partial sums of A013609 rows, first element of each row (equal to 1) excluded.

			Triangle:
n\k   1    2    3    4    5    6    7    8
--------------------------------------------
1     2
2     4    8
3     6   18   26
4     8   32   64   80
5    10   50  130  210  242
6    12   72  232  472  664  728
7    14   98  378  938 1610 2058 2186
8    16  128  576 1696 3488 5280 6304 6560
...
For instance, for n=3, in a cube:
k=1 corresponds to von Neumann's neighborhood with 6 neighbors situated on facets and given with offsets {(-1,0,0),(1,0,0),(0,-1,0),(0,1,0),(0,0,-1),(0,0,1)};
k=2 corresponds to 18 neighbors situated on facets and sides and given with offsets {(-1,0,0),(1,0,0),(0,-1,0),(0,1,0),(0,0,-1),(0,0,1),(-1,-1,0),(-1,0,-1),(0,-1,-1),(-1,0,1),(-1,1,0),(0,-1,1),(0,1,-1),(1,0,-1),(1,-1,0),(1,1,0),(1,0,1),(0,1,1)};
k=3 corresponds to Moore's neighborhood with 26 neighbors situated on facets, sides and corners given with offsets {(-1,0,0),(1,0,0),(0,-1,0),(0,1,0),(0,0,-1),(0,0,1),(-1,-1,0),(-1,0,-1),(0,-1,-1),(-1,0,1),(-1,1,0),(0,-1,1),(0,1,-1),(1,0,-1),(1,-1,0),(1,1,0),(1,0,1),(0,1,1),(-1,-1,-1),(1,-1,-1),(-1,1,-1),(1,1,-1),(-1,-1,1),(1,-1,1),(-1,1,1),(1,1,1)}.

D. A. Zaitsev, Generator of lattices
Dmitry Zaitsev, k-neighborhood for Cellular Automata, arXiv preprint arXiv:1605.08870 [cs.DM], 2016.
D. A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 666 (2017), 21-35.

First column equals to A005843.
Diagonal equals to A024023.
Partial row sums of A013609, first element of each row excluded.

T[n_, k_] := 3^n - 2^(k+1) Binomial[n, k+1] Hypergeometric2F1[1, k-n+1, k+2, -2] - 1;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 26 2018 *)

tabl(nn) = {for (n=1, nn, for (k=1, n, print1(sum(r=1, k, 2^r*binomial(n,r)), ", ");); print(););} \\ Michel Marcus, Dec 16 2015

T(n,k) = Sum_{r=1..k} 2^r*binomial(n,r).

Recurrence: T(n,k) = T(n-1,k-1)-2T(n-1,k-2)+T(n-1,k)+T(n,k-1), T(n,1) = 2n, T(n,n) = 3^n-1.

More terms from Michel Marcus, Dec 16 2015

A267849 Triangular array: T(n,k) is the 2-row creation rook number to place k rooks on a 3 x n board.

Original entry on oeis.org

1, 1, 3, 1, 6, 12, 1, 9, 36, 60, 1, 12, 72, 240, 360, 1, 15, 120, 600, 1800, 2520, 1, 18, 180, 1200, 5400, 15120, 20160, 1, 21, 252, 2100, 12600, 52920, 141120, 181440, 1, 24, 336, 3360, 25200, 141120, 564480, 1451520, 1814400, 1, 27, 432, 5040, 45360, 317520, 1693440, 6531840, 16329600, 19958400

Offset: 0

Ken Joffaniel M. Gonzales, Jan 21 2016

T(n,k) is the number of ways to place k rooks in a 3 x n Ferrers board (or diagram) under the Goldman-Haglund i-row creation rook mode for i=2. All row heights are 3.

			The triangle T(n,k) begins in row n=0 with columns 0<=k<=n:
     1
     1      3
     1      6     12
     1      9     36     60
     1     12     72    240    360
     1     15    120    600   1800   2520
     1     18    180   1200   5400  15120  20160
     1     21    252   2100  12600  52920 141120 181440
     1     24    336   3360  25200 141120 564480 1451520 1814400
     1     27    432   5040  45360 317520 1693440 6531840 16329600 19958400

Jay Goldman and James Haglund, Generalized rook polynomials, J. Combin. Theory A 91 (2000), 509-530.

Cf. A013610 (1-rook coefficients on the 3xn board), A121757 (2-rook coeffs. on the 2xn board), A013609 (1-rook coeffs. on the 2xn board), A013611 (1-rook coeffs. on the 4xn board), A008279 (2-rook coeffs. on the 1xn board), A082030 (row sums?), A049598 (column k=2), A007531 (column k=3 w/o factor 10), A001710 (diagonal?).

T(n,k) = T(n-1,k) + (k+2) T(n-1,k-1) subject to T(0,0)=1, T(n,k)=0 for n

Triangle simplified (reversing rows, offset 0). - R. J. Mathar, May 03 2017

A276985 Triangle read by rows: T(n,k) = number of k-dimensional elements in an n-dimensional cross-polytope, n>=1, 0<=k

Original entry on oeis.org

2, 4, 4, 6, 12, 8, 8, 24, 32, 16, 10, 40, 80, 80, 32, 12, 60, 160, 240, 192, 64, 14, 84, 280, 560, 672, 448, 128, 16, 112, 448, 1120, 1792, 1792, 1024, 256, 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120

Offset: 1

Felix Fröhlich, Sep 24 2016

It appears that this is 2*A193862 (but with a different offset) and that the sum of terms of the n-th row is A024023(n) = 3^n - 1. - Michel Marcus, Sep 29 2016

			T(4, 1..4) = 8, 24, 32, 16, because the 16-cell has 8 0-faces (vertices), 24 1-faces (edges), 32 2-faces (faces) and 16 3-faces (cells).
Triangle starts
2
4, 4
6, 12, 8
8, 24, 32, 16
10, 40, 80, 80, 32
12, 60, 160, 240, 192, 64
14, 84, 280, 560, 672, 448, 128
16, 112, 448, 1120, 1792, 1792, 1024, 256
18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512
20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024

H. S. M. Coxeter, Regular Polytopes, Third Edition, Dover Publications, 1973, ISBN 9780486141589.

Wikipedia, Cross-polytope.
D. A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 666 (2017), 21-35.

Cf. A038207 (hypercube), A135278 (simplex).
Rows: A005843(n), A046092(n), A130809(n+2), A130810(n+3).
Columns: A000079(n), A001787(n), A001788(n), A001789(n+3).
Cf. A013609, A024023, A182059, A193862.

Table[2^(k + 1) Binomial[n, k + 1], {n, 10}, {k, 0, n - 1}] // Flatten (* Michael De Vlieger, Sep 25 2016 *)

T(n, k) = 2^(k+1)*binomial(n, k+1)
trianglerows(n) = for(x=1, n, for(y=0, x-1, print1(T(x, y), ", ")); print(""))
trianglerows(10) \\ print initial 10 rows of triangle

T(n,k) = 2^(k+1) * binomial(n, k+1) (cf. Coxeter, 1973, formula 7.22).
T(n,k) = A182059(n,k) = A013609(n,k) . - R. J. Mathar, May 03 2017
G.f.: 2*x/((1 - x)*(1 - x - 2*x*y)). - Stefano Spezia, Jul 17 2025

A346911 Triangle read by rows: T(n,k) is the number of k-dimensional simplices with vertices from the n-dimensional cross polytope; 0 <= k < n.

Original entry on oeis.org

2, 4, 6, 6, 15, 8, 8, 28, 32, 16, 10, 45, 80, 80, 32, 12, 66, 160, 240, 192, 64, 14, 91, 280, 560, 672, 448, 128, 16, 120, 448, 1120, 1792, 1792, 1024, 256, 18, 153, 672, 2016, 4032, 5376, 4608, 2304, 512

Offset: 1

Peter Kagey, Aug 06 2021

			Table begins:
n\k |  0    1    2     3     4     5     6     7    8
----+-------------------------------------------------
  1 |  2
  2 |  4,   6
  3 |  6,  15,   8
  4 |  8,  28,  32,   16
  5 | 10,  45,  80,   80,   32
  6 | 12,  66, 160,  240,  192,   64
  7 | 14,  91, 280,  560,  672,  448,  128
  8 | 16, 120, 448, 1120, 1792, 1792, 1024,  256
  9 | 18, 153, 672, 2016, 4032, 5376, 4608, 2304, 512
Three of the T(3,1) = 15 1-simplices (line segments) in the 3-dimensional cross-polytope have vertices {(1,0,0), (-1,0,0)}, {(1,0,0), (0,1,0)}, and {(0,1,0), (0,0,-1)}.
One of the T(5,3) = 80 of the 3-simplices (tetrahedra) in the 5-dimensional cross-polytope has vertices {(1,0,0,0,0), (0,0,1,0,0), (0,0,0,-1,0), (0,0,0,0,1)}.

Cf. A013609, A346905.

T(n,0) = 2*n;
T(n,1) = 2*n^2-n;
T(n,k) = A013609(n,k+1) when k > 1.

A370469 Triangle read by columns where T(n,k) is the number of points in Z^n such that |x1| + ... + |xn| = k, |x1|, ..., |xn| > 0.

Original entry on oeis.org

2, 2, 4, 2, 8, 8, 2, 12, 24, 16, 2, 16, 48, 64, 32, 2, 20, 80, 160, 160, 64, 2, 24, 120, 320, 480, 384, 128, 2, 28, 168, 560, 1120, 1344, 896, 256, 2, 32, 224, 896, 2240, 3584, 3584, 2048, 512, 2, 36, 288, 1344, 4032, 8064, 10752, 9216, 4608, 1024

Offset: 1

Shel Kaphan, Mar 30 2024

T(n,k) is the number of points on the n-dimensional cross polytope with facets at distance k from the origin which have no coordinate equal to 0.
T(n,n) = 2^n. The (n-1)-dimensional simplex at distance n from the origin in Z^n has exactly 1 point with no zero coordinates, at (1,1,...,1). There are 2^n (n-1)-dimensional simplexes at distance n from the origin as part of the cross polytope in Z^n. (The lower dimensional facets do not count as they have at least one 0 coordinate.)
T(2*n,3*n) = T(2*n+1,3*n), and this is A036909.

			 n\k 1 2 3  4  5   6   7    8    9    10    11    12     13     14      15
   -----------------------------------------------------------------------
 1 | 2 2 2  2  2   2   2    2    2     2     2     2      2      2       2
 2 |   4 8 12 16  20  24   28   32    36    40    44     48     52      56
 3 |     8 24 48  80 120  168  224   288   360   440    528    624     728
 4 |       16 64 160 320  560  896  1344  1920  2640   3520   4576    5824
 5 |          32 160 480 1120 2240  4032  6720 10560  15840  22880   32032
 6 |              64 384 1344 3584  8064 16128 29568  50688  82368  128128
 7 |                 128  896 3584 10752 26880 59136 118272 219648  384384
 8 |                      256 2048  9216 30720 84480 202752 439296  878592
 9 |                           512  4608 23040 84480 253440 658944 1537536
10 |                                1024 10240 56320 225280 732160 2050048
11 |                                      2048 22528 135168 585728 2050048
12 |                                            4096  49152 319488 1490944
13 |                                                   8192 106496  745472
14 |                                                         16384  229376
15 |                                                                 32768
The cross polytope in Z^3 (the octahedron) with points at distance 3 from the origin has 8 triangle facets, each with edge length 4. There is one point in the center of each triangle with coordinates (+-1,+-1,+-1).

Cf. A033996, A333714 (n=3)
Cf. A102860 (n=4).
Cf. A036289, A097064, A134401 (+1-diagonal).
Cf. A001815 (+2-diagonal).
Cf. A371064.
Cf. A036909.
2 * A013609.

T[n_,k_]:=Binomial[k-1,n-1]*2^n; Table[T[n,k],{k,10},{n,k}]//Flatten

from math import comb
def A370469_T(n,k): return comb(k-1,n-1)<Chai Wah Wu, Apr 25 2024

T(n,k) = binomial(k-1,n-1)*2^n.

G.f.: 2*x*y/(1 - y - 2*x*y). - Stefano Spezia, Apr 27 2024

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A317501 Triangle read by rows: T(0,0) = 1; T(n,k) = 2*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.

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A350749 Triangle read by rows: T(n,k) is the number of oriented graphs on n labeled nodes with k arcs, n >= 0, k = 0..n*(n-1)/2.

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A191897 Coefficients of the Z(n,x) polynomials; Z(0,x) = 1, Z(1,x) = x and Z(n,x) = x*Z(n-1,x) - 2*Z(n-2,x), n >= 2.

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A265014 Triangle read by rows: T(n,k) = number of neighbors in n-dimensional lattice for generalized neighborhood given with parameter k.

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A267849 Triangular array: T(n,k) is the 2-row creation rook number to place k rooks on a 3 x n board.

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A276985 Triangle read by rows: T(n,k) = number of k-dimensional elements in an n-dimensional cross-polytope, n>=1, 0<=k

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A346911 Triangle read by rows: T(n,k) is the number of k-dimensional simplices with vertices from the n-dimensional cross polytope; 0 <= k < n.

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A370469 Triangle read by columns where T(n,k) is the number of points in Z^n such that |x1| + ... + |xn| = k, |x1|, ..., |xn| > 0.

A191897 Coefficients of the Z(n,x) polynomials; Z(0,x) = 1, Z(1,x) = x and Z(n,x) = xZ(n-1,x) - 2Z(n-2,x), n >= 2.