A130810 -id:A130810 - cp's OEIS frontend

A242153 Number T(n,k) of ascent sequences of length n with exactly k flat steps; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 16, 20, 12, 4, 1, 0, 61, 80, 50, 20, 5, 1, 0, 271, 366, 240, 100, 30, 6, 1, 0, 1372, 1897, 1281, 560, 175, 42, 7, 1, 0, 7795, 10976, 7588, 3416, 1120, 280, 56, 8, 1, 0, 49093, 70155, 49392, 22764, 7686, 2016, 420, 72, 9, 1, 0

Offset: 0

Joerg Arndt and Alois P. Heinz, May 05 2014

In general, column k is asymptotic to Pi^(2*k-5/2) / (k! * 6^(k-2) * sqrt(3) * exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014

			Triangle T(n,k) begins:
00:      1;
01:      1,     0;
02:      1,     1,     0;
03:      2,     2,     1,     0;
04:      5,     6,     3,     1,    0;
05:     16,    20,    12,     4,    1,    0;
06:     61,    80,    50,    20,    5,    1,   0;
07:    271,   366,   240,   100,   30,    6,   1,  0;
08:   1372,  1897,  1281,   560,  175,   42,   7,  1, 0;
09:   7795, 10976,  7588,  3416, 1120,  280,  56,  8, 1, 0;
10:  49093, 70155, 49392, 22764, 7686, 2016, 420, 72, 9, 1, 0;
...
The 15 ascent sequences of length 4 (dots denote zeros) with their number of flat steps are:
01:  [ . . . . ]   3
02:  [ . . . 1 ]   2
03:  [ . . 1 . ]   1
04:  [ . . 1 1 ]   2
05:  [ . . 1 2 ]   1
06:  [ . 1 . . ]   1
07:  [ . 1 . 1 ]   0
08:  [ . 1 . 2 ]   0
09:  [ . 1 1 . ]   1
10:  [ . 1 1 1 ]   2
11:  [ . 1 1 2 ]   1
12:  [ . 1 2 . ]   0
13:  [ . 1 2 1 ]   0
14:  [ . 1 2 2 ]   1
15:  [ . 1 2 3 ]   0
There are 5 sequences without flat steps, 6 with one flat step, etc., giving row [5, 6, 3, 1, 0] for n=4.

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A138265, A242154, A242155, A242156, A242157, A242158, A242159, A242160, A242161, A242162, A242163.
Row sums give A022493.
T(2n,n) gives A242164.
Main diagonal and lower diagonals give: A000007, A000012, A000027(n+1), A002378(n+1), A134481(n+1), A130810(n+4).
Cf. A137251 (the same for ascents), A238858 (the same for descents).

b:= proc(n, i, t) option remember; `if`(n=0, 1, expand(add(
      `if`(j=i, x, 1) *b(n-1, j, t+`if`(j>i, 1, 0)), j=0..t+1)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, -1$2)):
seq(T(n), n=0..12);

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Expand[Sum[If[j == i, x, 1]*b[n-1, j, t + If[j>i, 1, 0]], {j, 0, t+1}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][ b[n, -1, -1]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)

A130812 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 6-subsets of X containing none of X_i, (i=1,...n).

Original entry on oeis.org

64, 448, 1792, 5376, 13440, 29568, 59136, 109824, 192192, 320320, 512512, 792064, 1188096, 1736448, 2480640, 3472896, 4775232, 6460608, 8614144, 11334400, 14734720, 18944640, 24111360, 30401280, 38001600, 47121984, 57996288, 70884352, 86073856, 103882240

Offset: 6

Milan Janjic, Jul 16 2007

Number of n permutations (n>=6) of 3 objects u,v,z, with repetition allowed, containing n-6 u's. Example: if n=6 then n-6 =(0) zero u, a(1)=64. - Zerinvary Lajos, Aug 05 2008

a(n) is the number of 5-dimensional elements in an n-cross polytope where n>=6. - Patrick J. McNab, Jul 06 2015

Vincenzo Librandi, Table of n, a(n) for n = 6..1000
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. - Zerinvary Lajos, Aug 05 2008

[Binomial(2*n,6)+Binomial(n,2)*Binomial(2*n-4,2)- n*Binomial(2*n-2,4)-Binomial(n,3): n in [6..40]]; // Vincenzo Librandi, Jul 09 2015

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

CoefficientList[Series[64/(1-x)^7,{x,0,30}],x] (* Vincenzo Librandi, Mar 21 2012 *)

a(n) = binomial(2*n,6) + binomial(n,2)*binomial(2*n-4,2) - n*binomial(2*n-2,4) - binomial(n,3).
a(n) = C(n,n-6)*2^6, n>=6. - Zerinvary Lajos, Dec 07 2007
G.f.: 64*x^6/(1-x)^7. - Colin Barker, Mar 20 2012

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

A359662 Number of (3-dimensional) cells of regular m-polytopes for m >= 3.

Original entry on oeis.org

1, 5, 8, 15, 16, 24, 35, 40, 70, 80, 120, 126, 160, 210, 240, 330, 495, 560, 600, 715, 1001, 1120, 1365, 1792, 1820, 2016, 2380, 3060, 3360, 3876, 4845, 5280, 5376, 5985, 7315, 7920, 8855, 10626, 11440, 12650, 14950, 15360, 16016, 17550, 20475, 21840, 23751

Offset: 1

Marco Ripà, Jan 10 2023

In 3 dimensions there are five (convex) regular polytopes and each of them (trivially) consists of a single cell.
In 4 dimensions there are six regular 4-polytopes and they have 5, 8, 16, 24, 120, 600 3-dimensional cells (A063924).
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 (3-dimensional) cells in ascending order and define the present sequence.

			8 is a term since the hypersurface of a tesseract consists of 8 (cubical) cells.

MathStackExchange, MathStackExchange discussion concerning the number of edges of convex regular n-polytopes
Wikipedia, List of regular polytopes and compounds

Cf. A000332, A001789, A063924, A130810.

Cf. A359201 (edges), A359202 (faces).

Equals {{24, 120, 600} U {A000332} U {A001789} U {A130810}} \ {0}.

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

A373547 Triangle read by rows: T(n,k) = 4^k*binomial(n+k, n-k) with 0 <= k <= n.

Original entry on oeis.org

1, 1, 4, 1, 12, 16, 1, 24, 80, 64, 1, 40, 240, 448, 256, 1, 60, 560, 1792, 2304, 1024, 1, 84, 1120, 5376, 11520, 11264, 4096, 1, 112, 2016, 13440, 42240, 67584, 53248, 16384, 1, 144, 3360, 29568, 126720, 292864, 372736, 245760, 65536, 1, 180, 5280, 59136, 329472, 1025024, 1863680, 1966080, 1114112, 262144

Offset: 0

Stefano Spezia, Jun 09 2024

T(n,k) is the number of occurrences of the periodic substring (01)^k in the periodic string (0011)^n (see Proposition 4.3 at page 6 in Fang).
The word (w_1, w_2, ..., w_r)^m is defined as the word obtained by concatenating (w_1, w_2, ..., w_r) m times.
A word w' = (w'1, w'_2, ..., w'_s) is said be a subword of a given word w = (w_1, w_2, ..., w_r), if there is some set P = {p_1 < ... < p_s} of integers from 1 to r satisfying w{p_j} = w'_j for all 1 <= j <= s, and we call the set P an occurrence of w' in w (see Preliminaries section at pp. 2-3 in Fang).

			The triangle begins as:
  1;
  1,  4;
  1, 12,   16;
  1, 24,   80,   64;
  1, 40,  240,  448,   256;
  1, 60,  560, 1792,  2304,  1024;
  1, 84, 1120, 5376, 11520, 11264, 4096;
  ...
T(2,1) = 12 since there are 12 occurrences of (01)^1 = 01 in (0011)^2 = 00110011: {1, 3}, {1, 4}, {1, 7}, {1, 8}, {2, 3}, {2, 4}, {2, 7}, {2, 8}, {5, 7}, {5, 8}, {6, 7}, {6, 8}.

Wenjie Fang, Maximal number of subword occurrences in a word, arXiv:2406.02971 [math.CO], 2024.

Cf. A000012 (k=0), A000302 (diagonal), A001653 (row sums), A046092 (k=1), A046717, A085478, A130810, A130812, A373628.

T[n_,k_]:=4^k Binomial[n+k,n-k]; Table[T[n,k],{n,0,9},{k,0,n}]//Flatten (* or *)
T[n_,k_]:=SeriesCoefficient[(1-x)/((1-x)^2-4x y),{x,0,n},{y,0,k}]; Table[T[n,k],{n,0,9},{k,0,n}]//Flatten

G.f.: (1 - x)/((1 - x)^2 - 4*x*y).
T(n,k) = A000302(k)*A085478(n,k).
Sum_{k=0..n} T(n-k,k) = A046717(n).
T(n,2) = A130810(n+2).
T(n,3) = A130812(n+3).

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

A242153 Number T(n,k) of ascent sequences of length n with exactly k flat steps; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A130812 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 6-subsets of X containing none of X_i, (i=1,...n).

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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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A359662 Number of (3-dimensional) cells of regular m-polytopes for m >= 3.

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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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A373547 Triangle read by rows: T(n,k) = 4^k*binomial(n+k, n-k) with 0 <= k <= n.

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