A001788 -id:A001788 - cp's OEIS frontend

A320906 T(n, k) = binomial(2n - k, k - 1)hypergeom([2, 2, 1 - k], [1, 2*(1 - k + n)], -1), triangle read by rows, T(n,k) for n >= 0 and 0 <= k <= n.

0, 0, 1, 0, 1, 6, 0, 1, 8, 24, 0, 1, 10, 39, 80, 0, 1, 12, 58, 150, 240, 0, 1, 14, 81, 256, 501, 672, 0, 1, 16, 108, 406, 955, 1524, 1792, 0, 1, 18, 139, 608, 1686, 3178, 4339, 4608, 0, 1, 20, 174, 870, 2794, 6144, 9740, 11762, 11520

Offset: 0

Peter Luschny, Oct 28 2018

			Triangle starts:
[0] 0
[1] 0, 1
[2] 0, 1,  6
[3] 0, 1,  8,  24
[4] 0, 1, 10,  39,  80
[5] 0, 1, 12,  58, 150,  240
[6] 0, 1, 14,  81, 256,  501,  672
[7] 0, 1, 16, 108, 406,  955, 1524, 1792
[8] 0, 1, 18, 139, 608, 1686, 3178, 4339,  4608
[9] 0, 1, 20, 174, 870, 2794, 6144, 9740, 11762, 11520

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A320907. T(n, n) = A001788(n).

Cf. A320905.

T := (n, k) -> binomial(2*n-k, k-1)*hypergeom([2, 2, 1-k], [1, 2*(1-k+n)], -1):
seq(seq(simplify(T(n, k)), k=0..n), n=0..9);

T[n_, k_] := Sum[Binomial[2*n+1-k, 2*n+2-2*k+j]*Binomial[j+2, 2], {j,0, 2*n+1-k}]; Flatten[Table[T[n, k], {n, 0, 15}, {k, 0, n}]] (* Detlef Meya, Dec 31 2023 *)

T(n, k) = {sum(j=0, 2*n+1-k, binomial(2*n+1-k, 2*n+2-2*k+j) * binomial(j+2,2))} \\ Andrew Howroyd, Dec 31 2023

from functools import cache
@cache
def T(n, k):
    if k <= 0 or n <= 0: return 0
    if k == 1: return 1
    if k == n: return n * (n + 1) * 2**(n - 2)
    return T(n-1, k) + 2*T(n-1, k-1) - T(n-2, k-2)
for n in range(10): print([T(n, k) for k in range(n + 1)])
# after Detlef Meya, Peter Luschny, Jan 01 2024

T(n, k) = Sum_{j=0..2*n+1-k} binomial(2*n+1-k, 2*n+2-2*k+j) * binomial(j+2,2). - Detlef Meya, Dec 31 2023

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}.

A372868 Irregular triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of weak ascents, with 1 <= k <= ceiling(n/2).

Original entry on oeis.org

1, 2, 4, 1, 8, 6, 16, 24, 1, 32, 80, 10, 64, 240, 60, 1, 128, 672, 280, 14, 256, 1792, 1120, 112, 1, 512, 4608, 4032, 672, 18, 1024, 11520, 13440, 3360, 180, 1, 2048, 28160, 42240, 14784, 1320, 22, 4096, 67584, 126720, 59136, 7920, 264, 1, 8192, 159744, 366080, 219648, 41184, 2288, 26

Offset: 1

Stefano Spezia, May 15 2024

With offset 0 for the variable k, T(n,k) is the number of flattened Catalan words of length n with exactly k peaks. In such case, T(4,1) = 6 corresponds to 6 flattened Catalan words of length 4 with 1 peak: 0010, 0100, 0110, 0101, 0120, and 0121. See Baril et al. at page 20.

			The irregular triangle begins:
    1;
    2;
    4,    1;
    8,    6;
   16,   24,    1;
   32,   80,   10;
   64,  240,   60,   1;
  128,  672,  280,  14;
  256, 1792, 1120, 112, 1;
  ...
T(4,2) = 6 since there are 6 flattened Catalan words of length 4 with 2 runs of weak ascents: 0010, 0100, 0101, 0110, 0120, and 0121.

Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See pp. 8-9.

Cf. A000079, A001788, A002409, A003472, A007051 (row sums), A110654 (row lengths), A140325, A172242.

Cf. A372852, A372972.

T[n_,k_]:=SeriesCoefficient[(1-2x)*x*y/(1-4*x+4*x^2-x^2*y),{x,0,n},{y,0,k}]; Table[T[n,k],{n,14},{k,Ceiling[n/2]}] //Flatten (* or *)
T[n_,k_]:=2^(n-2k+1)Binomial[n-1,2k-2]; Table[T[n,k],{n,14},{k,Ceiling[n/2]}]

G.f.: (1-2*x)*x*y/(1 - 4*x + 4*x^2 - x^2*y).
T(n,k) = 2^(n-2*k+1)*binomial(n-1, 2*k-2).
T(n,1) = A000079(n-1).
T(n,2) = A001788(n-2).
T(n,3) = A003472(n-1).
T(n,4) = A002409(n-7).
T(n,5) = A140325(n-9).
T(n,6) = A172242(n-1).
Sum_{k>=0} T(n,k) = A007051(n-1).

A381899 Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1} such that I(x) + W(x)*(n-W(x)) = k, where I(x) is the number of inversions in x and W(x) is the number of 1's in x, n >= 0, 0 <= k <= floor(n^2/2).

Original entry on oeis.org

1, 2, 2, 1, 1, 2, 0, 2, 2, 2, 2, 0, 0, 2, 3, 3, 4, 1, 1, 2, 0, 0, 0, 2, 2, 4, 4, 6, 4, 4, 2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 4, 5, 7, 6, 9, 7, 7, 5, 4, 1, 1, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 4, 4, 8, 6, 10, 12, 14, 12, 14, 10, 10, 6, 4, 2, 2

Offset: 0

Geoffrey Critzer, Mar 09 2025

Sum_{k>=0} T(n,k)*2^k = A132186(n).
Sum_{k>=0} T(n,k)*3^k = A053846(n).
Sum_{k>=0} T(n,k)*q^k = the number of idempotent n X n matrices over GF(q).
It appears that if n is even the n-th row converges to 2,0,0,...,21,13,9,5,4,1,1 which is A226622 reversed, and if n is odd the sequence is twice A226635.
From Alois P. Heinz, Mar 09 2025: (Start)
Sum_{k>=0} k * T(n,k) = 3*A001788(n-1) for n>=1.
Sum_{k>=0} (-1)^k * T(n,k) = A060546(n). (End)

			Triangle T(n,k) begins:
  1;
  2;
  2, 1, 1;
  2, 0, 2, 2, 2;
  2, 0, 0, 2, 3, 3, 4, 1, 1;
  2, 0, 0, 0, 2, 2, 4, 4, 6, 4, 4, 2, 2;
  ...
T(4,5) = 3 because we have: {0, 1, 0, 0}, {0, 1, 0, 1}, {1, 1, 0, 1}.

Alois P. Heinz, Rows n = 0..50, flattened

Row sums give A000079.

Cf. A001788, A060546, A226635, A226622, A132186, A053846, A083906.

b:= proc(i, j) option remember; expand(`if`(i+j=0, 1,
     `if`(i=0, 0, b(i-1, j))+`if`(j=0, 0, b(i, j-1)*z^i)))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(
         expand(add(b(n-j, j)*z^(j*(n-j)), j=0..n))):
seq(T(n), n=0..10);  # Alois P. Heinz, Mar 09 2025

nn = 7; B[n_] := FunctionExpand[QFactorial[n, q]]*q^Binomial[n, 2];e[z_] := Sum[z^n/B[n], {n, 0, nn}]; Map[CoefficientList[#, q] &, Table[B[n], {n, 0, nn}] CoefficientList[Series[e[z]^2, {z, 0, nn}],z]]

Sum_{n>=0} Sum_{k>=0} T(n,k)*q^k*x^n/(n_q!*q^binomial(n,2)) = e(x)^2 where e(x) = Sum_{n>=0} x^n/(n_q!*q^binomial(n,2)) where n_q! = Product{i=1..n} (q^n-1)/(q-1).

A085408 Total number of cycles in the binary n-cube.

Original entry on oeis.org

0, 1, 28, 14704, 51109385408

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 12 2003

Computed by Daniele Degiorgi (danieled(AT)INF.ETHZ.CH).

Richard H. Hammack, Paul C. Kainen, Graph Bases and Diagram Commutativity, Graphs and Combinatorics (2018), Vol. 34, Issue 4, 523-534.
Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Hypercube Graph

Cf. A066037, A001788. Row sums of A085452.

Table[Total[Table[Length[FindCycle[HypercubeGraph[n], {k}, All]], {k, 4, 2^n, 2}]], {n, 4}] (* Eric W. Weisstein, Mar 24 2020 *)

A102841 a(n) = ((9n^2 + 33n + 26)*2^n + (-1)^n)/27.

Original entry on oeis.org

1, 5, 19, 61, 179, 493, 1299, 3309, 8211, 19949, 47635, 112109, 260627, 599533, 1366547, 3089901, 6937107, 15476205, 34331155, 75769325, 166451731, 364127725, 793500179, 1723082221, 3729512979, 8048092653, 17319057939

Offset: 0

Creighton Dement, Feb 27 2005

nonn

A floretion-generated sequence relating the number of edges and faces in n-dimensional hypercube.
Equals A001787, (1, 4, 12, 32, 80, ...) convolved with A001045, the Jacobsthal sequence. - Gary W. Adamson, May 23 2009
The sum of the sizes of all inversions in compositions of n. - Arnold Knopfmacher, Jan 22 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
Index entries for linear recurrences with constant coefficients, signature (5,-6,-4,8).

Cf. A045883, A001788, A001793, A102301.

Cf. A001787, A001045. - Gary W. Adamson, May 23 2009

[((9*n^2 + 33*n + 26)*2^n + (-1)^n)/27 : n in [0..40]]; // Wesley Ivan Hurt, Jul 03 2020

Table[(1/27)*((9 n^2 + 33 n + 26) 2^n + (-1)^n), {n, 0, 50}] (* or *) LinearRecurrence[{5,-6,-4,8}, {1,5,19,61}, 50] (* G. C. Greubel, Sep 27 2017 *)

G.f.: 1/((1+x)*(1-2*x)^3).
a(n+1) - 2*a(n) = A045883(n+2).
a(n) + a(n+1) = A001788(n+2).
a(n) = 5*a(n-1) - 6*a(n-2) - 4*a(n-3) + 8*a(n-4). - Wesley Ivan Hurt, Jul 03 2020

Corrected by T. D. Noe, Nov 08 2006

A130749 Triangle A007318*A090181 (as infinite lower triangular matrices) .

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 24, 10, 1, 1, 31, 80, 60, 15, 1, 1, 63, 240, 280, 125, 21, 1, 1, 127, 672, 1120, 770, 231, 28, 1, 1, 255, 1792, 4032, 3920, 1806, 392, 36, 1, 1, 511, 4608, 13440, 17472, 11340, 3780, 624, 45, 1

Offset: 0

Philippe Deléham, Jul 13 2007

			Triangle begins:
  1;
  1,   1;
  1,   3,    1;
  1,   7,    6,     1;
  1,  15,   24,    10,     1;
  1,  31,   80,    60,    15,     1;
  1,  63,  240,   280,   125,    21,    1;
  1, 127,  672,  1120,   770,   231,   28,   1;
  1, 255, 1792,  4032,  3920,  1806,  392,  36,  1;
  1, 511, 4608, 13440, 17472, 11340, 3780, 624, 45,  1;
  ...

Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Sherry H. F. Yan, Schroeder Paths and Pattern Avoiding Partitions, arXiv:0805.2465 [math.CO], 2008-2009; Corollary 3.6.

Cf. A000012, A000225, A001788, A003472 ; A000012, A000217, A014205.

nmax = 9;
T1[n_, k_] := Binomial[n, k];
T2[n_, k_] := Sum[(-1)^(j-k) Binomial[2n-j, j] Binomial[j, k] CatalanNumber[n-j], {j, 0, n}];
T[n_, k_] := Sum[T1[n, m] T2[m, k], {m, 0, n}];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)

N(n, k):=(binomial(n, k-1)*binomial(n, k))/n;
T(n, k):=if k=0 then 1 else sum(binomial(n, i)*N(i, k), i, 1, n); /* Vladimir Kruchinin, Jan 08 2022 */

Sum_{k=0..n} T(n,k) = A007317(n+1).
G.f.: 1/(1-x-xy/(1-x/(1-x-xy/(1-x/(1-x-xy/(1-x.... (continued fraction); [Paul Barry, Jan 12 2009]
T(n,k) = Sum_{i=1..n} binomial(n, i)*N(i,k), T(n,0)=1, where N(n,k) is the triangle of Narayana numbers A001263. - Vladimir Kruchinin, Jan 08 2022

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

A171631 Triangle read by rows: T(n,k) = n(binomial(n-2, k-1) + nbinomial(n-2, k)), n > 0 and 0 <= k <= n - 1.

Original entry on oeis.org

1, 4, 2, 9, 12, 3, 16, 36, 24, 4, 25, 80, 90, 40, 5, 36, 150, 240, 180, 60, 6, 49, 252, 525, 560, 315, 84, 7, 64, 392, 1008, 1400, 1120, 504, 112, 8, 81, 576, 1764, 3024, 3150, 2016, 756, 144, 9, 100, 810, 2880, 5880, 7560, 6300, 3360, 1080, 180, 10, 121, 1100

Offset: 1

Roger L. Bagula, Dec 13 2009

If T(0,0) = 0 is prepended, then row sums give A001788.

			Triangle begins:
n\k|  0    1     2     3     4     6    7    8  9
-------------------------------------------------
1  |  1
2  |  4    2
3  |  9   12     3
4  | 16   36    24     4
5  | 25   80    90    40     5
6  | 36  150   240   180    60     6
7  | 49  252   525   560   315    84    7
8  | 64  392  1008  1400  1120   504  112    8
9  | 81  576  1764  3024  3150  2016  756  144  9
... reformatted. - _Franck Maminirina Ramaharo_, Oct 02 2018

Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Publications, 1945, p. 32.

Cf. A003506, A007318, A127952, A171531.

Table[CoefficientList[n*(x + n)*(x + 1)^(n - 2), x], {n, 1, 12}]//Flatten

T(n, k) := n*(binomial(n - 2, k - 1) + n*binomial(n - 2, k))$
tabl(nn) := for n:1 thru nn do print(makelist(T(n, k), k, 0, n - 1))$ /* Franck Maminirina Ramaharo, Oct 02 2018 */

Let p(x;n) = (x + 1)^n. Then row n are the coefficients in the expansion of p''(x;n) - x*p'(x;n) + n*p(x;n) = n*(x + n)*(x + 1)^(n - 2).
From Franck Maminirina Ramaharo, Oct 02 2018: (Start)
T(n,1) = A000290(n).
T(n,2) = A011379(n).
T(n,3) = 3*A002417(n-2).
T(n,n-2) = A046092(n-1).
T(n,n-3) = 9*A000292(n-2).
G.f.: y*(x*y - y - 1)/(x*y + y - 1)^3. (End)

Edited and new name by Franck Maminirina Ramaharo, Oct 02 2018

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

cp's OEIS Frontend

A320906 T(n, k) = binomial(2*n - k, k - 1)*hypergeom([2, 2, 1 - k], [1, 2*(1 - k + n)], -1), triangle read by rows, T(n,k) for n >= 0 and 0 <= k <= n.

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

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A372868 Irregular triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of weak ascents, with 1 <= k <= ceiling(n/2).

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A381899 Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1} such that I(x) + W(x)*(n-W(x)) = k, where I(x) is the number of inversions in x and W(x) is the number of 1's in x, n >= 0, 0 <= k <= floor(n^2/2).

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Maple

Mathematica

Formula

A085408 Total number of cycles in the binary n-cube.

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Mathematica

A102841 a(n) = ((9*n^2 + 33*n + 26)*2^n + (-1)^n)/27.

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Magma

Mathematica

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Extensions

A130749 Triangle A007318*A090181 (as infinite lower triangular matrices) .

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Mathematica

Maxima

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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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A320906 T(n, k) = binomial(2n - k, k - 1)hypergeom([2, 2, 1 - k], [1, 2*(1 - k + n)], -1), triangle read by rows, T(n,k) for n >= 0 and 0 <= k <= n.

A102841 a(n) = ((9n^2 + 33n + 26)*2^n + (-1)^n)/27.

A171631 Triangle read by rows: T(n,k) = n(binomial(n-2, k-1) + nbinomial(n-2, k)), n > 0 and 0 <= k <= n - 1.