A255982 -id:A255982 - cp's OEIS frontend

A001761 a(n) = (2*n)!/(n+1)!.

1, 1, 4, 30, 336, 5040, 95040, 2162160, 57657600, 1764322560, 60949324800, 2346549004800, 99638080819200, 4626053752320000, 233153109116928000, 12677700308232960000, 739781100339240960000, 46113021921146019840000

Offset: 0

N. J. A. Sloane

nonn

According to the Beineke and Pippert paper, the number of dissections of a disk is given by D(n)=R(n)/(n-2)!, where R(n)=A001761(n-2) is the number of labeled planar 2-trees having n vertices and rooted at a given exterior edge. [Clarified by M. F. Hasler, Feb 22 2012]
a(n+1) is the number of labeled incomplete ternary trees on n vertices in which each left and middle child have a larger label than their parent. - Brian Drake, Jul 28 2008
For n>0: a(n) = A173333(2*n,n+1); cf. A006963, A001813. - Reinhard Zumkeller, Feb 19 2010

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, pp. 12-26 of Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970. Reprinted in Math. Annalen, Vol. 191 (1971), pp. 87-98.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Peter J. Cameron, Some treelike objects, Quart. J. Math. Oxford, Vol. 38, No. 2 (1987), pp. 155-183. See p. 166. - _N. J. A. Sloane_, Apr 18 2014
Ali Chouria, Vlad-Florin Drǎgoi, and Jean-Gabriel Luque, On recursively defined combinatorial classes and labelled trees, arXiv:2004.04203 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 80.
K. A. Penson and J.-M. Sixdeniers, Integral Representations of Catalan and Related Numbers, J. Integer Sequences, Vol. 4 (2001), Article 01.2.5.
Karol A. Penson and Allan I. Solomon, Coherent states from combinatorial sequences, arXiv:quant-ph/0111151, 2001.

Cf. A000108, A173333, A006963, A001813.

Main diagonal of A255982, A256061.

seq(mul((n+k), k=2..n), n=0..17); # Zerinvary Lajos, Feb 15 2008

Table[(2*n)!/(n+1)!,{n,0,20}] (* Vincenzo Librandi, Feb 23 2012 *)

combinat::catalan(n)*n! $ n = 0..17; // Zerinvary Lajos, Feb 15 2007

A001761(n)=binomial(2*n,n+1)*(n-1)!  \\ M. F. Hasler, Feb 23 2012

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=sum(k=0,n,(-1)^(n-k)*(n+1)^(k-1)*Stirling1(n,k))} \\ Paul D. Hanna, Nov 09 2012

[binomial(2*n,n)/(1+n)*factorial(n) for n in range(0, 18)] # Zerinvary Lajos, Dec 03 2009

a(n) = n!*Catalan(n) =n!* A000108(n). - N. J. A. Sloane, Apr 18 2014
a(n+2) = sum(A038455(n, m), m=1..n), n >= 1. - Wolfdieter Lang
E.g.f. for this sequence = o.g.f. for A000108. - Len Smiley, Dec 07 2001
Integral representation as the moment of a positive function on the positive half-axis: in Maple notation, a(n)=int(x^n*(-1/2+exp(-x/4)/sqrt(Pi*x)+erf(sqrt(x)/2)/2), x=0..infinity), n=0, 1... This representation is unique. - Karol A. Penson, Aug 21 2001
G.f.: If G_N(x)=1+sum('(2*k)!*(x^k)/(k+1)!', 'k'=1..N),  G_N(x)=1+2*x/(G(0)-2*x); G(k)=4*x*(k^2)+6*k*x+k+2*x+2-2*x*(2*k+3)*((k+2)^2)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
a(n) = Sum_{k=0..n} (-1)^(n-k) * (n+1)^(k-1) * Stirling1(n,k). - Paul D. Hanna, Nov 09 2012
G.f.: Q(0) where Q(k) =  1 + x*(2*k+1)*(4*k+1)/(k+1 - 4*x*(k+1)^2*(4*k+3)/(4*x*(k+1)*(4*k+3) + (2*k+3)/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 05 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x/(x + (k+2)/(2*k+2)/(2*k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013
Let A(x) = sum(k>=0, a(k)*x^k /(2*k)! ) = ( exp(x)-1)/x, then A(x) = 1/Q(0), where Q(k) = 1 - x/( 1 + (2*k+1)/(1 - x/( 1 + 2*(k+1)/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2013
From Ilya Gutkovskiy, Jan 21 2017: (Start)
a(n) ~ sqrt(2)*4^n*n^(n-1)/exp(n).
Sum_{n>=0} 1/a(n) = (7*exp(1/4)*sqrt(Pi)*erf(1/2) + 10)/8 = 2.2865189388213215..., where erf() is the error function. (End)
D-finite with recurrence: (n+1)*a(n) -2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Feb 16 2020
Sum_{n>=0} (-1)^n/a(n) = 3/4 - 5*sqrt(Pi)*erfi(1/2)/(8*exp(1/4)), where erfi() is the imaginary error function. - Amiram Eldar, Apr 03 2022

A237018 Number A(n,k) of partitions of the k-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 8, 5, 0, 1, 4, 18, 39, 14, 0, 1, 5, 32, 132, 212, 42, 0, 1, 6, 50, 314, 1080, 1232, 132, 0, 1, 7, 72, 615, 3440, 9450, 7492, 429, 0, 1, 8, 98, 1065, 8450, 40320, 86544, 47082, 1430, 0, 1, 9, 128, 1694, 17604, 124250, 494736, 819154, 303336, 4862, 0

Offset: 0

Alois P. Heinz, Feb 02 2014

The g.f. given below is a generalization of formulas given by Murray R. Bremner and Sara Madariaga in A236339 and A236342. According to them A(n,k) also gives the number of distinct monomials of degree n+1 in the universal algebra with k nonassociative binary products {*1,...,*k} related only by the interchange laws from k-category theory: (a *i b) *j (c *i d) = (a *j c) *i (b *j d) for i,j in {1,...,k} and i

These numbers can be regarded as (one of many possible definitions of) higher-dimensional Catalan numbers. - N. J. A. Sloane, Feb 12 2014

			A(3,1) = 5:
  [||-|---], [-|||---], [-|-|-|-], [---|||-], [---|-||].
  .
A(2,2) = 8:
  ._______.  ._______.  ._______.  ._______.
  | | |   |  |   | | |  |_______|  |       |
  | | |   |  |   | | |  |_______|  |_______|
  | | |   |  |   | | |  |       |  |_______|
  |_|_|___|  |___|_|_|  |_______|  |_______|
  ._______.  ._______.  ._______.  ._______.
  |   |   |  |   |   |  |   |   |  |       |
  |___|   |  |   |___|  |___|___|  |_______|
  |   |   |  |   |   |  |       |  |   |   |
  |___|___|  |___|___|  |_______|  |___|___|.
  .
Square array A(n,k) begins:
  1,   1,    1,     1,      1,       1,       1, ...
  0,   1,    2,     3,      4,       5,       6, ...
  0,   2,    8,    18,     32,      50,      72, ...
  0,   5,   39,   132,    314,     615,    1065, ...
  0,  14,  212,  1080,   3440,    8450,   17604, ...
  0,  42, 1232,  9450,  40320,  124250,  311472, ...
  0, 132, 7492, 86544, 494736, 1912900, 5770692, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Yu Hin (Gary) Au, Fatemeh Bagherzadeh, Murray R. Bremner, Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube, arXiv:1903.00813 [math.CO], Mar 03 2019, p. 8.

Columns k=0-10 give: A000007, A000108, A236339(n+1), A236342(n+1), A237019, A237020, A237021, A237022, A237023, A237024, A237025.
Rows n=0-2 give: A000012, A001477, A001105.
Main diagonal gives A237026.
Cf. A255982.

A:= (n, k)-> coeff(series(RootOf(x*(-1)^k=add((-1)^i*
    binomial(k, i)*(G*x)^(2^(k-i)), i=0..k), G), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
       A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, k_, t_] := b[n, k, t] = If[t == 0, 1, If[t == 1, A[n-1, k], Sum[A[j, k]*b[n-j-1, k, t-1], {j, 0, n-2}]]]; A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[ Binomial[k, j]*(-1)^j*b[n+1, k, 2^j], {j, 1, k}]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

G.f. G_k of column k satisfies: (-1)^k*x = Sum_{i=0..k} (-1)^i*C(k,i)*(G_k*x)^(2^(k-i)).

A(n,k) = Sum_{i=0..k} C(k,i) * A255982(n,i). - Alois P. Heinz, Mar 13 2015

A256061 Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 4, 0, 5, 30, 30, 0, 14, 196, 504, 336, 0, 42, 1260, 6300, 10080, 5040, 0, 132, 8184, 71280, 205920, 237600, 95040, 0, 429, 54054, 774774, 3603600, 7207200, 6486480, 2162160, 0, 1430, 363220, 8288280, 58378320, 180180000, 273873600, 201801600, 57657600

Offset: 0

Alois P. Heinz, Mar 13 2015

Also number of binary trees with n inner nodes of exactly k different dimensions.  T(2,2) = 4:
: balanced parentheses :  ([]) :  [()] : ()[]  : []()  :
:----------------------:-------:-------:-------:-------:
:                trees :   (1) :   [2] : (1)   : [2]   :
:                      :   / \ :   / \ : / \   : / \   :
:                      : [2]   : (1)   :   [2] :   (1) :
:                      : / \   : / \   :   / \ :   / \ :

			A(3,2) = 30: (())[], (()[]), (([])), ()()[], ()([]), ()[()], ()[[]], ()[](), ()[][], ([()]), ([[]]), ([]()), ([])(), ([])[], ([][]), [(())], [()()], [()[]], [()](), [()][], [([])], [[()]], [[]()], [[]](), [](()), []()(), []()[], []([]), [][()], [][]().
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,     4;
  0,   5,    30,     30;
  0,  14,   196,    504,     336;
  0,  42,  1260,   6300,   10080,    5040;
  0, 132,  8184,  71280,  205920,  237600,   95040;
  0, 429, 54054, 774774, 3603600, 7207200, 6486480, 2162160;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-1 give: A000007, A000108 (for n>0).
Main diagonal gives A001761.
Cf. A253180, A255982, A258427, A290605.

ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

A[0, 0] = 1; A[n_, k_] := A[n, k] = k^n*CatalanNumber[n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 20 2017, translated from Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * (k-i)^n * A000108(n).
T(n,k) = k! * A253180(n,k).
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A290605(n,k-i). - Alois P. Heinz, Oct 28 2019

A258426 Number of partitions of the n-dimensional hypercube resulting from a sequence of 2n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.

Original entry on oeis.org

1, 2, 184, 64464, 51622600, 74699100720, 171052924578480, 569565504689176800, 2601107886874207253760, 15609810973119409265234400, 119149819949135773678717267200, 1127426871984268618976053945104000, 12953029027945569352833762868999449600

Offset: 0

Alois P. Heinz, May 29 2015

nonn

			a(1) = 2 : [||-],  [-||].

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..139 (terms 0..70 from Alois P. Heinz)

Cf. A255982, A258399.

b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
       A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
    end:
T:= proc(n, k) option remember;
      add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
    end:
a:= n-> T(2*n,n):
seq(a(n), n=0..15);

b[n_, k_, t_] := b[n, k, t] = If[t == 0, 1, If[t == 1, A[n - 1, k], Sum[A[j, k]*b[n - j - 1, k, t - 1], {j, 0, n - 2}]]]; A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[Binomial[k, j]*(-1)^j*b[n + 1, k, 2^j], {j, 1, k}]]; T[n_, k_] := T[n, k] = Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}]; a[n_] := T[2*n, n]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

a(n) = A255982(2n,n).

a(n) ~ c * d^n * n!^2 / n^(5/2), where d = A256254 = 98.8248737517356857317..., c = 2^(3/8) * (-LambertW(-2*exp(-2)))^(1/8) / (8 * Pi^(3/2) * sqrt(1 + LambertW(-2*exp(-2)))) = 0.033762267258894908009578351704834892... . - Vaclav Kotesovec, May 31 2015, updated Sep 27 2023

A258427 Number T(n,k) of redundant binary trees with n inner nodes of exactly k different dimensions used for the partition of the k-dimensional hypercube by hierarchical bisection; triangle T(n,k), n>=3, 2<=k<=n-1, read by rows.

Original entry on oeis.org

1, 12, 18, 112, 420, 336, 956, 6816, 12936, 7200, 7830, 95579, 324540, 414450, 178200, 62744, 1244466, 6755720, 14886300, 14355000, 5045040, 496518, 15537456, 127063596, 430572780, 699460740, 542341800, 161441280

Offset: 3

Alois P. Heinz, May 29 2015

T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. T(n,k) = 0 for k in {0, 1} or k>=n.

			T(3,2) = 1. There are A256061(3,2) = 30 binary trees with 3 inner nodes of exactly 2 different dimensions, 28 of them have unique hypercube partitions, 2 of them have the same partition:
:              :                     : partition :
|--------------|---------------------|-----------|
|              |    (1)       [2]    |           |
|              |    / \       / \    |   .___.   |
|       trees: |  [2] [2]   (1) (1)  |   |_|_|   |
|              |  / \ / \   / \ / \  |   |_|_|   |
|    balanced  |                     |           |
| parentheses: |  ([])[]    [()]()   |           |
|--------------|---------------------|-----------|
Triangle T(n,k) begins:
.
. .
. .     .
. .     1,       .
. .    12,      18,       .
. .   112,     420,     336,        .
. .   956,    6816,   12936,     7200,        .
. .  7830,   95579,  324540,   414450,   178200,       .
. . 62744, 1244466, 6755720, 14886300, 14355000, 5045040,   .

Alois P. Heinz, Rows n = 3..135, flattened

Cf. A256061, A255982.

A:= proc(n, k) option remember; k^n*binomial(2*n, n)/(n+1) end:
B:= proc(n, k) option remember;
       add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
    end:
b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
       H(n-1, k), add(H(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
    end:
H:= proc(n, k) option remember; `if`(n=0, 1,
      -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
    end:
G:= proc(n, k) option remember;
       add(H(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
    end:
T:= (n, k)-> B(n, k)-G(n, k):
seq(seq(T(n, k), k=2..n-1), n=3..12);

A[n_, k_] := A[n, k] = k^n*Binomial[2*n, n]/(n+1); B[n_, k_] := B[n, k] = Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; b[n_, k_, t_] := b[n, k, t] = If[t==0, 1, If[t==1, H[n-1, k], Sum[H[j, k]*b[n-j-1, k, t-1], {j, 0, n-2}]]]; H[n_, k_] := H[n, k] = If[n==0, 1, -Sum[Binomial[k, j]* (-1)^j* b[n+1, k, 2^j], {j, 1, k}]]; G[n_, k_] := G[n, k] = Sum[H[n, k-i]*(-1)^i* Binomial[k, i], {i, 0, k}]; T[n_, k_] := T[n, k] = B[n, k]-G[n, k]; Table[Table[T[n, k], {k, 2, n-1}], {n, 3, 12}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

T(n,k) = A256061(n,k) - A255982(n,k).

A258416 Number of partitions of the 2-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.

Original entry on oeis.org

4, 29, 184, 1148, 7228, 46224, 300476, 1983102, 13266032, 89795420, 614058228, 4236652416, 29457698192, 206215486597, 1452248529432, 10281676045348, 73137772914324, 522472109334560, 3746685545297640, 26961148855455180, 194626321451800800, 1409026233004925340

Offset: 2

Alois P. Heinz, May 29 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Column k=2 of A255982.

b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
       A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
    end:
T:= proc(n, k) option remember;
      add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
    end:
a:= n-> T(n, 2):
seq(a(n), n=2..25);

b[n_, k_, t_] := b[n, k, t] = If[t == 0, 1, If[t == 1, A[n - 1, k], Sum[A[j, k]*b[n - j - 1, k, t - 1], {j, 0, n - 2}]]];
A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[Binomial[k, j]*(-1)^j*b[n + 1, k, 2^j], {j, 1, k}]];
T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
a[n_] := T[n, 2];
a /@ Range[2, 25] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

From Vaclav Kotesovec, May 29 2015: (Start)
Recurrence: 5*(n-2)*(n-1)*n*(n+1)*(13616*n^4 - 138092*n^3 + 514558*n^2 - 835288*n + 498441)*a(n) =  - 6*(n-2)*(n-1)*n*(27232*n^5 - 289800*n^4 + 1170888*n^3 - 2195854*n^2 + 1802270*n - 411881)*a(n-1) + 16*(n-2)*(n-1)*(544640*n^6 - 6612960*n^5 + 32102192*n^4 - 79406652*n^3 + 104891690*n^2 - 69498516*n + 17766135)*a(n-2) - 8*(n-2)*(2*n - 5)*(1524992*n^6 - 18516288*n^5 + 89869136*n^4 - 222469596*n^3 + 295082666*n^2 - 197989116*n + 52268391)*a(n-3) - 16*(2*n - 7)*(2*n - 5)*(4*n - 13)*(4*n - 11)*(13616*n^4 - 83628*n^3 + 181978*n^2 - 165984*n + 53235)*a(n-4).
a(n) ~ c * d^n / (sqrt(Pi) * n^(3/2)), where d = 7.721133226857077553917531558... is the root of the equation 256 + 512*d - 32*d^2 - 5*d^3 = 0, c = 1.11097484883257916279675191289... is the root of the equation -8 + 364*c^2 - 518*c^4 + 185*c^6 = 0.
(End)

A258417 Number of partitions of the 3-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.

Original entry on oeis.org

30, 486, 5880, 64464, 679195, 7043814, 72707844, 751082244, 7785793080, 81092511276, 849060054420, 8937364804760, 94564644817767, 1005496779910572, 10740560345206680, 115218669255806304, 1240869923563291014, 13412271463669969704, 145454088924589697192

Offset: 3

Alois P. Heinz, May 29 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Vaclav Kotesovec, Recurrence (of order 11)

Column k=3 of A255982.

b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
       A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
    end:
T:= proc(n, k) option remember;
      add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
    end:
a:= n-> T(n, 3):
seq(a(n), n=3..25);

b[n_, k_, t_] := b[n, k, t] = If[t == 0, 1, If[t == 1, A[n - 1, k], Sum[A[j, k]*b[n - j - 1, k, t - 1], {j, 0, n - 2}]]];
A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[Binomial[k, j]*(-1)^j*b[n + 1, k, 2^j], {j, 1, k}]];
T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
a[n_] := T[n, 3];
a /@ Range[3, 25] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = 11.6335027253872064795086728699206569842475549795979388187955249065144... is the root of the equation 16777216 - 150994944*d + 1716387840*d^3 + 2063339520*d^4 - 6994944*d^5 - 21019200*d^6 + 454313*d^7 = 0 and c = 0.6170954330535517584816422123448632671500498041324155957832713069267... . - Vaclav Kotesovec, Feb 20 2016

A258418 Number of partitions of the 4-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.

Original entry on oeis.org

336, 9744, 192984, 3279060, 51622600, 779602164, 11499880768, 167393051696, 2419080596520, 34838703973728, 501182126787744, 7212689238965297, 103937431212291680, 1500609318117978064, 21713411768745550544, 314940143510352714144, 4579270473409470432352

Offset: 4

Alois P. Heinz, May 29 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..800

Column k=4 of A255982.

b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
       A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
    end:
T:= proc(n, k) option remember;
      add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
    end:
a:= n-> T(n, 4):
seq(a(n), n=4..25);

b[n_, k_, t_] := b[n, k, t] = If[t == 0, 1, If[t == 1, A[n - 1, k], Sum[A[j, k]*b[n - j - 1, k, t - 1], {j, 0, n - 2}]]];
A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[Binomial[k, j]*(-1)^j*b[n + 1, k, 2^j], {j, 1, k}]];
T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
a[n_] := T[n, 4];
a /@ Range[4, 25] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

A258419 Number of partitions of the 5-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.

Original entry on oeis.org

5040, 230400, 6792750, 165293700, 3624918660, 74699100720, 1479942440340, 28577108044800, 542482698531000, 10181610525525360, 189663357076785270, 3515970161266821510, 64985380300281057950, 1199146771516702098500, 22111945264260791498090

Offset: 5

Alois P. Heinz, May 29 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..750

Column k=5 of A255982.

b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
       A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
    end:
T:= proc(n, k) option remember;
      add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
    end:
a:= n-> T(n, 5):
seq(a(n), n=5..25);

b[n_, k_, t_] := b[n, k, t] = If[t == 0, 1, If[t == 1, A[n - 1, k], Sum[A[j, k]*b[n - j - 1, k, t - 1], {j, 0, n - 2}]]];
A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[Binomial[k, j]*(-1)^j*b[n + 1, k, 2^j], {j, 1, k}]];
T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
a[n_] := T[n, 5];
a /@ Range[5, 25] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

A258420 Number of partitions of the 6-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.

Original entry on oeis.org

95040, 6308280, 259518600, 8563232700, 249224561040, 6703099068120, 171052924578480, 4209175565848800, 100941470303368480, 2376150752752629210, 55182874193888254800, 1268931845185709426820, 28968880808493233206500, 657875495503038733415880

Offset: 6

Alois P. Heinz, May 29 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..700

Column k=6 of A255982.

b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
       A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
    end:
T:= proc(n, k) option remember;
      add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
    end:
a:= n-> T(n, 6):
seq(a(n), n=6..25);

Showing 1-10 of 15 results. Next

cp's OEIS Frontend

A001761 a(n) = (2*n)!/(n+1)!.

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A237018 Number A(n,k) of partitions of the k-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A256061 Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A258426 Number of partitions of the n-dimensional hypercube resulting from a sequence of 2n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.

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A258427 Number T(n,k) of redundant binary trees with n inner nodes of exactly k different dimensions used for the partition of the k-dimensional hypercube by hierarchical bisection; triangle T(n,k), n>=3, 2<=k<=n-1, read by rows.

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A258416 Number of partitions of the 2-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.

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A258417 Number of partitions of the 3-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.

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