A130809 -id:A130809 - cp's OEIS frontend

A319577 a(n) = (4/45)n(n - 2)(n - 1)(n^3 - 12n^2 + 47n - 15).

0, 0, 0, 24, 96, 240, 544, 1288, 3136, 7392, 16320, 33528, 64416, 116688, 200928, 331240, 525952, 808384, 1207680, 1759704, 2508000, 3504816, 4812192, 6503112, 8662720, 11389600, 14797120, 19014840, 24189984, 30488976, 38099040, 47229864, 58115328, 71015296

Offset: 0

Peter Luschny, Oct 01 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000012 (m=0), A005843 (m=1), A046092 (m=2), A130809 (m=3), A319575 (m=4), A319576 (m=5), this sequence (m=6).
Column n=6 of A122141.
Cf. A319574.

a := n -> (4/45)*n*(n - 2)*(n - 1)*(n^3 - 12*n^2 + 47*n - 15):
seq(a(n), n=0..41);

A319577[n_]:=4/45*n*(n-2)*(n-1)*(n^3-12*n^2+47*n-15); Array[A319577, 50, 0] (*or*)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 0, 24, 96, 240, 544}, 50] (* Paolo Xausa, Feb 20 2024 *)

concat([0,0,0], Vec(8*x^3*(3 - 9*x + 9*x^2 + 5*x^3) / (1 - x)^7 + O(x^40))) \\ Colin Barker, Oct 02 2018

a(n) = [x^6] JacobiTheta3(x)^n.
a(n) = A319574(n,6).
From Colin Barker, Oct 02 2018: (Start)
G.f.: 8*x^3*(3 - 9*x + 9*x^2 + 5*x^3) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6.
(End)

A130811 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 5-subsets of X containing none of X_i, (i=1,...n).

Original entry on oeis.org

32, 192, 672, 1792, 4032, 8064, 14784, 25344, 41184, 64064, 96096, 139776, 198016, 274176, 372096, 496128, 651168, 842688, 1076768, 1360128, 1700160, 2104960, 2583360, 3144960, 3800160, 4560192, 5437152, 6444032, 7594752, 8904192

Offset: 5

Milan Janjic, Jul 16 2007

Number of n permutations (n>=5) of 3 objects u,v,z, with repetition allowed, containing n-5 u's. Example: if n=5 then n-5 =(0) zero u, a(1)=32. - Zerinvary Lajos, Aug 05 2008

a(n) is the number of 4-dimensional elements in an n-cross polytope where n>=5. - Patrick J. McNab, Jul 06 2015

Milan Janjic, Two Enumerative Functions
Eric Weisstein's World of Mathematics, Cross Polytope

Cf. A038207, A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A046092, A130809, A130810. - Zerinvary Lajos, Aug 05 2008

[Binomial(n,n-5)*2^5: n in [5..40]]; // Vincenzo Librandi, Jul 09 2015

a:=n->binomial(2*n,5)+(2*n-4)*binomial(n,2)-n*binomial(2*n-2,3)
seq(binomial(n,n-5)*2^5,n=5..34); # Zerinvary Lajos, Dec 07 2007
seq(binomial(n+4, 5)*2^5, n=1..22); # Zerinvary Lajos, Aug 05 2008

Table[Binomial[2 n, 5] + (2 n - 4) Binomial[n, 2] - n Binomial[2 n - 2, 3], {n, 5, 40}] (* Vincenzo Librandi, Jul 09 2015 *)

a(n) = binomial(2*n,5) + (2*n-4)*binomial(n,2) - n*binomial(2*n-2,3).
a(n) = C(n,n-5)*2^5, for n>=5. - Zerinvary Lajos, Dec 07 2007
G.f.: 32*x^5/(1-x)^6. - Colin Barker, Apr 14 2012

A359202 Number of (bidimensional) faces of regular m-polytopes for m >= 3.

Original entry on oeis.org

4, 6, 8, 10, 12, 20, 24, 32, 35, 56, 80, 84, 96, 120, 160, 165, 220, 240, 280, 286, 364, 448, 455, 560, 672, 680, 720, 816, 960, 969, 1140, 1200, 1320, 1330, 1540, 1760, 1771, 1792, 2024, 2288, 2300, 2600, 2912, 2925, 3276, 3640, 3654, 4060, 4480, 4495, 4608

Offset: 1

Marco Ripà, Dec 20 2022

In 3 dimensions there are five (convex) regular polytopes and they have 4, 6, 8, 12, or 20 (bidimensional) faces (A053016).
In 4 dimensions there are six regular 4-polytopes and they have 10, 24, 32, 96, 720, or 1200 faces (A063925).
In m >= 5 dimensions, there are only 3 regular polytopes (i.e., the m-simplex, the m-cube, and the m-crosspolytope) so that we can sort their number of bidimensional faces in ascending order and define the present sequence.

			6 is a term since a cube has 6 faces.

Mathematics StackExchange, What are the formulas for the number of vertices, edges, faces, cells, 4-faces, ..., n-faces, of convex regular polytopes in n≥5 dimensions?
Wikipedia, List of regular polytopes and compounds

Cf. A000292, A001788, A053016, A063925, A130809.

Cf. A359201 (edges), A359662 (cells).

{a(n), n >= 1} = {{12, 96, 720, 1200} U {A000292} U {A001788} U {A130809}} \ {0, 1}.

A360604 Triangle read by rows. T(n, k) = 2^binomial(n - k, 2) * binomial(n - 1, k - 1).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 8, 6, 3, 1, 0, 64, 32, 12, 4, 1, 0, 1024, 320, 80, 20, 5, 1, 0, 32768, 6144, 960, 160, 30, 6, 1, 0, 2097152, 229376, 21504, 2240, 280, 42, 7, 1, 0, 268435456, 16777216, 917504, 57344, 4480, 448, 56, 8, 1

Offset: 0

Peter Luschny, Feb 23 2023

			Triangle T(n, k) starts:
[0] 1;
[1] 0,       1;
[2] 0,       1,      1;
[3] 0,       2,      2,     1;
[4] 0,       8,      6,     3,    1;
[5] 0,      64,     32,    12,    4,   1;
[6] 0,    1024,    320,    80,   20,   5,  1;
[7] 0,   32768,   6144,   960,  160,  30,  6, 1;
[8] 0, 2097152, 229376, 21504, 2240, 280, 42, 7, 1;

Cf. A006125 (column 1), A002378 (T(n+2,n)), A130809 (T(n+3,n)), A006896 (row sums).

T := (n, k) -> 2^binomial(n - k, 2) * binomial(n-1, k-1):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

A371532 Centered cuboctahedral numbers: the number of integer triples (x,y,z) such that max(|x|,|y|,|z|) <= n and |x|+|y|+|z| <= 2n.

Original entry on oeis.org

1, 19, 93, 263, 569, 1051, 1749, 2703, 3953, 5539, 7501, 9879, 12713, 16043, 19909, 24351, 29409, 35123, 41533, 48679, 56601, 65339, 74933, 85423, 96849, 109251, 122669, 137143, 152713, 169419, 187301, 206399, 226753, 248403, 271389, 295751, 321529, 348763

Offset: 0

Peter Kagey, Mar 26 2024

			The a(1) = 19 lattice points are all permutations of the points (0,0,0), (0,0,1), and (0,1,1), where any number of the coordinates can also be made negative (e.g., (1,-1,0)).

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Cuboctahedron.
Eric Weisstein's World of Mathematics, Figurate Number.
Wikipedia, Ehrhart polynomial.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A016755, A130809, A371515.

Array[(20*#^3 + 24*#^2 + 10*# + 3)/3 &, 50, 0] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {1, 19, 93, 263}, 50] (* Paolo Xausa, Apr 02 2024 *)

def A371532(n): return n*(n*(5*n+6<<2)+10)//3+1 # Chai Wah Wu, Apr 02 2024

a(n) = (20*n^3 + 24*n^2 + 10*n + 3)/3.
a(n) =  A016755(n) - A130809(n-2).
G.f.: (x^3 + 23*x^2 + 15*x + 1)/(x-1)^4. - Paolo Xausa, Apr 02 2024
From Elmo R. Oliveira, Aug 22 2025: (Start)
E.g.f.: exp(x)*(3 + 54*x + 84*x^2 + 20*x^3)/3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

A130813 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 7-subsets of X containing none of X_i, (i=1,...n).

Original entry on oeis.org

128, 1024, 4608, 15360, 42240, 101376, 219648, 439296, 823680, 1464320, 2489344, 4073472, 6449664, 9922560, 14883840, 21829632, 31380096, 44301312, 61529600, 84198400, 113667840, 151557120, 199779840, 260582400, 336585600, 430829568

Offset: 7

Milan Janjic, Jul 16 2007

nonn

Number of n permutations (n>=7) of 3 objects u,v,z, with repetition allowed, containing n-7 u's. Example: if n=7 then n-7 =(0) zero u, a(1)=128. - Zerinvary Lajos, Aug 05 2008

a(n) is the number of 6-dimensional elements in an n-cross polytope where n>=7. - Patrick J. McNab, Jul 06 2015

Milan Janjic, Two Enumerative Functions
Eric Weisstein's World of Mathematics, Cross Polytope

Cf. A038207, A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A046092, A130809, A130810, A130811, A130812. - Zerinvary Lajos, Aug 05 2008

Cf. A000580.

[Binomial(n,n-7)*2^7: n in [7..40]]; // Vincenzo Librandi, Jul 09 2015

a:=n->binomial(2*n,7)+binomial(n,2)*binomial(2*n-4,3)-n*binomial(2*n-2,5)-(2*n-6)*binomial(n,3);
seq(binomial(n,n-7)*2^7,n=7..32); # Zerinvary Lajos, Dec 07 2007
seq(binomial(n+6, 7)*2^7, n=1..22); # Zerinvary Lajos, Aug 05 2008

Table[Binomial[n, n - 7] 2^7, {n, 7, 40}] (* Vincenzo Librandi, Jul 09 2015 *)

a(n) = binomial(2*n,7) + binomial(n,2)*binomial(2*n-4,3) - n*binomial(2*n-2,5) - (2*n-6)*binomial(n,3).
a(n) = C(n,n-7)*2^7, n>=7. - Zerinvary Lajos, Dec 07 2007
G.f.: 128*x^7/(1-x)^8. - Colin Barker, Mar 18 2012
a(n) = 128*A000580(n). a(n+1) = 2*(n+1)*a(n)/(n-6) for n >= 7. - Robert Israel, Jul 08 2015

A185342 Triangle of successive recurrences in columns of A117317(n).

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

Offset: 0

Paul Curtz, Jan 26 2012

A117317 (A):
1
2    1
4    5   1
8   16   9   1
16  44  41  14   1
32 112 146  85  20  1
64 272 456 377 155 27 1
have for their columns successive signatures
(2) (4,-4) (6,-12,8) (8,-24, 32, -16) (10,-40,80,-80,32) i.e. a(n).
Take based on abs(A133156) (B):
1
2   0
4   1  0
8   4  0 0
16 12  1 0 0
32 32  6 0 0 0
64 80 24 1 0 0 0.
The recurrences of successive columns are also a(n). a(n) columns: A005843(n+1), A046092(n+1), A130809, A130810, A130811, A130812, A130813.
A053220 + A001787 = A014480.

			Triangle T(n,k),for 1<=k<=n, begins :
2                                         (1)
4    -4                                   (2)
6   -12   8                               (3)
8   -24  32   -16                         (4)
10  -40  80   -80   32                    (5)
12  -60 160  -240  192   -64              (6)
14  -84 280  -560  672  -448  128         (7)
16 -112 448 -1120 1792 -1792 1024 -256    (8)
Successive rows can be divided by A171977.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. For (A): A053220, A056243. For (B): A000079, A001787, A001788, A001789. For A193862: A115068 (a Coxeter group). For (2): A014480 (also (3),(4),(5),..); also A053220 and A001787.

Cf. A007318.

Table[(-1)*Binomial[n, k]*(-2)^k, {n, 1, 20}, {k, 1, n}] // Flatten (* G. C. Greubel, Jun 27 2017 *)

for(n=1,20, for(k=1,n, print1((-2)^(k+1)*binomial(n,k)/2, ", "))) \\ G. C. Greubel, Jun 27 2017

Take A133156(n) without 1's or -1's double triangle (C)=
2
4
8      -4
16    -12
32    -32   6
64    -80  24
128  -192  80   -8
256  -448 240  -40
512 -1024 672 -160 10;
a(n) is increasing odd diagonals and increasing (sign changed) even diagonals. Rows sum of (C) = A201629 (?) Another link between Chebyshev polynomials and cos( ).
Absolute values: A013609(n) without 1's. Also 2*A193862 = (2*A002260)*A135278.
T(n,k) = T(n-1,k) - 2*T(n-1,k-1) for k>1, T(n,1) = 2*n = 2*T(n-1,1) - T(n-2,1). - Philippe Deléham, Feb 11 2012
T(n,k) = (-1)* Binomial(n,k)*(-2)^k, 1<=k<=n. - Philippe Deléham, Feb 11 2012

A206022 Riordan array (1, xexp(arcsinh(-2x))).

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 2, -4, 1, 0, 0, 8, -6, 1, 0, -2, -8, 18, -8, 1, 0, 0, 0, -32, 32, -10, 1, 0, 4, 8, 30, -80, 50, -12, 1, 0, 0, 0, 0, 128, -160, 72, -14, 1, 0, -10, -16, -28, -112, 350, -280, 98, -16, 1, 0, 0, 0

Offset: 0

Philippe Deléham, Feb 02 2012

Riordan array (1, x*(sqrt(1+4x^2)-2x)); inverse of Riordan array (1, x/sqrt(1-4x)), see A205813.
The g.f. for row sums (1,1,-1,-1,3,1,-9,1,27,13,-81,67,243,...) is (1+2*x^2+x*sqrt(1+4*x^2))/(1+3*x^2).
Triangle T(n,k), read by rows, given by (0, -2, 1, -1, 1, -1, 1, -1, 1, -1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

			Triangle begins:
  1
  0,   1
  0,  -2,   1
  0,   2,  -4,   1
  0,   0,   8,  -6,    1,
  0,  -2,  -8,  18,   -8,    1
  0,   0,   0, -32,   32,  -10,     1
  0,   4,   8,  30,  -80,   50,   -12,    1
  0,   0,   0,   0,  128, -160,    72,  -14,    1
  0, -10, -16, -28, -112,  350,  -280,   98,  -16,   1
  0,   0,   0,   0,    0, -512,   768, -448,  128, -18,   1
  0,  28,  40,  54,   96,  420, -1512, 1470, -672, 162, -20, 1

Cf. A104624 (column k=1).

T(n,n) = 1, T(n+1,n) = -2n = -A005843(n), T(n+2,n) = 2*n^2 = A001105(n), T(n+3,n) = -A130809(n+1), T(2n,n) = A009117(n), T(2n+3,1) = (-1)^n*2*A000108(n).

T(n,k) = T(n-2,k-2) - 4*T(n-2,k-1), for k >= 2.

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

cp's OEIS Frontend

A319577 a(n) = (4/45)*n*(n - 2)*(n - 1)*(n^3 - 12*n^2 + 47*n - 15).

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A130811 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 5-subsets of X containing none of X_i, (i=1,...n).

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A359202 Number of (bidimensional) faces of regular m-polytopes for m >= 3.

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A360604 Triangle read by rows. T(n, k) = 2^binomial(n - k, 2) * binomial(n - 1, k - 1).

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A371532 Centered cuboctahedral numbers: the number of integer triples (x,y,z) such that max(|x|,|y|,|z|) <= n and |x|+|y|+|z| <= 2n.

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A130813 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 7-subsets of X containing none of X_i, (i=1,...n).

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A185342 Triangle of successive recurrences in columns of A117317(n).

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A206022 Riordan array (1, x*exp(arcsinh(-2*x))).

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A319577 a(n) = (4/45)n(n - 2)(n - 1)(n^3 - 12n^2 + 47n - 15).

A206022 Riordan array (1, xexp(arcsinh(-2x))).