A236339 -id:A236339 - cp's OEIS frontend

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.

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

A372233 Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-x-x^2) )^n.

Original entry on oeis.org

1, 2, 12, 77, 520, 3612, 25557, 183192, 1325808, 9666635, 70897112, 522472392, 3865669717, 28697325048, 213649228560, 1594540806612, 11926354293792, 89372808145692, 670865679851667, 5043360211505000, 37965778448487120, 286151354441445570, 2159143860124095120

Offset: 0

Seiichi Manyama, May 02 2024

nonn

Cf. A213684, A236339, A372458, A372460.

A372233 := proc(n)
    add(binomial(n+k-1,k) * binomial(3*n-k-1,n-2*k),k=0..floor(n/2));
end proc:
seq(A372233(n),n=0..50) ; # R. J. Mathar, May 02 2024

Table[SeriesCoefficient[1/((1-x)*(1-x-x^2))^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 04 2024 *)

a(n, s=2, t=1, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t+u+1)*n-(s-1)*k-1, n-s*k));

a(n) = Sum_{k=0..floor(n/2)} binomial(n+k-1,k) * binomial(3*n-k-1,n-2*k).
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x) * (1-x-x^2) ).
D-finite with recurrence +575*n*(n-1)*(n-2)*a(n) +40*(n-1)*(n-2)*(125*n-178)*a(n-1) -16*(n-2)*(3272*n^2-5536*n+75)*a(n-2) +8*(-22112*n^3+169392*n^2-450082*n+415827)*a(n-3) +1344*(96*n^3-1328*n^2+5794*n-8139)*a(n-4) +3072*(4*n-15)*(2*n-9)*(4*n-17)*a(n-5)=0. - R. J. Mathar, May 02 2024
a(n) ~ sqrt((1/8 + cos(arccos(sqrt(37)/8)/3)/sqrt(37))/(Pi*n)) / (-2/3 + sqrt(35/18)*cos(arccos(-4537/(560*sqrt(70)))/3))^n. - Vaclav Kotesovec, May 04 2024

A368931 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^3) ).

Original entry on oeis.org

1, 2, 7, 31, 154, 819, 4560, 26244, 154874, 932074, 5698745, 35297535, 221016593, 1396717756, 8896798020, 57062237502, 368201804973, 2388587515239, 15568995139404, 101913055166811, 669678357109300, 4415837460391845, 29210203356645090

Offset: 0

Seiichi Manyama, Jan 10 2024

nonn

Cf. A236339, A368932, A368933.

Cf. A049140, A054514.

a(n) = sum(k=0, n\3, binomial(n+k, k)*binomial(3*n-2*k+1, n-3*k))/(n+1);

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*(1-x-x^3))/x)

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+k,k) * binomial(3*n-2*k+1,n-3*k).

A236342 Association types in 3-dimensional algebra.

Original entry on oeis.org

1, 3, 18, 132, 1080, 9450, 86544, 819154, 7949532, 78671736, 790930728, 8055355698, 82935309996, 861772240368, 9025745922656, 95183320362093, 1009853631571878, 10771405762277094, 115438084007465376, 1242437345193084264, 13423511539998223884

Offset: 1

Murray R. Bremner, Jan 22 2014

This sequence has two equivalent descriptions:
(1) It enumerates the number of decompositions of the unit cube into n rectangular parallelepipeds obtained by the following algorithm.
(a) Start with the unit cube.
(b) Perform the following operation n-1 times: Choose a parallelepiped in the current decomposition. Bisect this parallelepiped into two parallelepipeds by a plane orthogonal to any of the 3 coordinate axes. Different sequences of bisections can produce the same decomposition.
(2) Consider the universal algebra with three nonassociative binary products *1, *2, *3 related only by the three interchange laws from 2-category theory, as follows where (i,j) = (1,2), (1,3), (2,3):
   ( a *i b ) *j ( c *i d ) = ( a *j c ) *i ( b *i d )
This sequence enumerates the number of distinct monomials of degree n.

J.-L. Loday and B. Vallette, Algebraic Operads, Grundlehren 346, Springer, 2012, section 13.10.4, page 544 (for the interchange law).
S. Mac Lane, Categories for the Working Mathematician, second edition, Springer, 1978, equation (5), page 43 (also for the interchange law).

Alois P. Heinz, Table of n, a(n) for n = 1..400
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.
Murray Bremner, Sara Madariaga, Permutation of elements in double semigroups, arXiv:1405.2889 [math.RA], 2014-2015.
Murray Bremner, Sara Madariaga, Permutation of elements in double semigroups, Semigroup Forum 92 (2016), no. 2, 335--360. MR3472020.
Wikipedia, Parallelepiped

Cf. A000108 (for 1-dimensional algebra), A236339 (for 2-dimensional algebra).

Column k=3 of A237018.

MAXDEG := 24:
C[ 1 ] := 1:
for n from 2 to MAXDEG do
  count := 0:
  for k to 3 do
     count := count +
     ( (-1)^(k-1) * binomial(3,k) *
     add( mul( C[f], f in e ), e in combinat[composition](n,2^k) ) )
  od:
  print( n, count ):
  C[ n ] := count
od:

Rest[CoefficientList[InverseSeries[Series[-x^8+3*x^4-3*x^2+x, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Feb 16 2014 *)

Recurrence relation:
C(1) = 1,
C(n) = 3 sum_{i1,i2} C(i1)C(i2)
   - 3 sum_{i1,i2,i3,i4} C(i1)C(i2)C(i3)C(i4)
   + sum_{i1,i2,i3,i4,i5,i6,i7,i8} C(i1)C(i2)C(i3)C(i4)C(i5)C(i6)C(i7)C(i8).
The first sum is over all 2-compositions of n into positive integers, the second sum is over all 4-compositions, and the third sum is over all 8-compositions.
This recurrence relation has a natural generalization using inclusion-exclusion to k-dimensional algebras for all k > 0, where k = 1 gives the familiar classical Catalan numbers, but with offset 1 not the usual offset 0; that is, k = 1 has the n-th term 1/n*binomial(2*n-2,n-1) instead of the more familiar 1/(n+1)*binomial(2*n,n) (thanks to Alois P. Heinz for pointing this out).
Generating function: G(x) = sum_{n>=1} C(n)x^n satisfies a polynomial of degree 8: G(x)^8 - 3G(x)^4 + 3G(x)^2 - G(x) + x = 0.
a(n) ~ (1/r)^(n-1/2) / (sqrt(2*Pi*(6-36*s^2+56*s^6)) * n^(3/2)), where s = 0.17792425007438691... is the root of the equation 8*s^7-12*s^3+6*s = 1, and r = s*(7-18*s+12*s^3)/8 = 0.085958633749898... - Vaclav Kotesovec, Feb 16 2014

A300474 Number of partitions of the square resulting from a sequence of n n-sections, each of which divides any part perpendicular to any of the axes.

Original entry on oeis.org

1, 1, 8, 96, 2240, 80960, 4021248, 255704064, 19878918144, 1829788646400, 194788537180160, 23556611967336448, 3191162612827078656, 478807179615908462592, 78833945248222913495040, 14133035289273287214366720, 2740751307013005651817267200

Offset: 0

Alois P. Heinz, Dec 15 2018

nonn

			a(2) = 8:
  ._______.  ._______.  ._______.  ._______.
  | | |   |  |   | | |  |_______|  |       |
  | | |   |  |   | | |  |_______|  |_______|
  | | |   |  |   | | |  |       |  |_______|
  |_|_|___|  |___|_|_|  |_______|  |_______|
  ._______.  ._______.  ._______.  ._______.
  |   |   |  |   |   |  |   |   |  |       |
  |___|   |  |   |___|  |___|___|  |_______|
  |   |   |  |   |   |  |       |  |   |   |
  |___|___|  |___|___|  |_______|  |___|___|.
  .

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A091144, A236339, A237026, A300613, A322543.

a:= proc(n) option remember; `if`(n<2, 1, coeff(series(
      RootOf(G-x-2*G^n+G^(n^2), G), x, n^2-n+2), x, n^2-n+1))
    end:
seq(a(n), n=0..16);

a[0] = a[1] = 1; a[n_] := Module[{G}, G[] = 0; Do[G[x] = 2 G[x]^n - G[x]^n^2 + x + O[x]^(n^2 - n + 2) // Normal, {n^2 - n + 2}];
Coefficient[G[x], x, n^2 - n + 1]];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Dec 29 2018, after Alois P. Heinz *)

A368932 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^4) ).

Original entry on oeis.org

1, 2, 7, 30, 144, 741, 3996, 22287, 127495, 743941, 4410555, 26492349, 160875186, 986007700, 6091548256, 37894543413, 237168491610, 1492323419929, 9434943086870, 59906035386393, 381832957589226, 2442251022673595, 15670578495195870

Offset: 0

Seiichi Manyama, Jan 10 2024

nonn

Cf. A236339, A368931, A368933.

Cf. A063021, A215342.

a(n) = sum(k=0, n\4, binomial(n+k, k)*binomial(3*n-3*k+1, n-4*k))/(n+1);

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*(1-x-x^4))/x)

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} binomial(n+k,k) * binomial(3*n-3*k+1,n-4*k).

A368933 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^5) ).

Original entry on oeis.org

1, 2, 7, 30, 143, 729, 3891, 21471, 121505, 701316, 4112751, 24435424, 146773582, 889813460, 5437598036, 33459382065, 207138653334, 1289231982454, 8062548100445, 50637167131635, 319255808742145, 2019867936975125, 12819928874057325, 81603361510347675

Offset: 0

Seiichi Manyama, Jan 10 2024

nonn

Cf. A236339, A368931, A368932.

Cf. A276989, A366112.

a(n) = sum(k=0, n\5, binomial(n+k, k)*binomial(3*n-4*k+1, n-5*k))/(n+1);

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*(1-x-x^5))/x)

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/5)} binomial(n+k,k) * binomial(3*n-4*k+1,n-5*k).

A322543 Number of triadic partitions of the unit square into (2n+1) subrectangles.

Original entry on oeis.org

1, 2, 12, 96, 879, 8712, 90972, 985728, 10979577, 124937892, 1446119664, 16972881120, 201526230555, 2416309004872, 29215072931136, 355800894005760, 4360705642282569, 53744080256387478, 665667989498682936, 8281518339078928800, 103441301833577854041, 1296713265300164761632

Offset: 0

Yu Hin Au, Dec 14 2018

nonn

A kind of two-dimensional ternary Catalan number. This sequence enumerates the decompositions of the unit square into 2n+1 rectangles obtained by the following algorithm.
(a) Start with the unit square.
(b) Perform the following operation n times:
    (1) Choose a rectangle in the current decomposition.
    (2) Trisect this rectangle into three rectangles horizontally or vertically.
Note that different sequences of trisections can produce the same decomposition.

Alois P. Heinz, Table of n, a(n) for n = 0..889
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.

Cf. A000108 (Catalan numbers), A005408, A236339 (decompositions of unit square using bisections).

a:= n-> coeff(series(RootOf(G^9-2*G^3+G-x, G), x, 2*n+2), x, 2*n+1):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 14 2018

a[n_] := SeriesCoefficient[InverseSeries[x - 2 x^3 + x^9 + O[x]^(2n+2), x], {x, 0, 2n+1}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 13 2019, from PARI *)

a(n)={polcoef(serreverse(x - 2*x^3 + x^9 + O(x^(2*n+2))), 2*n+1)} \\ Andrew Howroyd, Dec 14 2018

Recurrence relation: a(n) = C(2n+1) with C(1) = 1 and C(n) = 2 Sum_{i1,i2,i3} C(i1)C(i2)C(i3) - Sum_{i1,i2,i3,i4,i5,i6,i7,i8,i9} C(i1)C(i2)C(i3)C(i4)C(i5)C(i6)C(i7)C(i8)C(i9). The first sum is over all 3-compositions of n into positive integers (i1+i2+i3=n), and the second sum is over all 9-compositions of n into positive integers (i1+i2+...+i9=n).

a(n) = [x^(2n+1)] G(x), where G(x) satisfies: G(x)^9 - 2*G(x)^3 + G(x) - x = 0.

A368934 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x-x^2) ).

Original entry on oeis.org

1, 3, 16, 104, 751, 5789, 46656, 388377, 3313304, 28816513, 254548840, 2277498340, 20596833817, 187974816142, 1729033498416, 16012809644088, 149186508912927, 1397300099214753, 13149137686976324, 124262625068365924, 1178796712807563025

Offset: 0

Seiichi Manyama, Jan 10 2024

nonn

Cf. A001002, A046736, A236339.

Cf. A368938.

a(n) = sum(k=0, n\2, binomial(n+k, k)*binomial(4*n-k+2, n-2*k))/(n+1);

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2*(1-x-x^2))/x)

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(4*n-k+2,n-2*k).

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A129442 Expansion of c(x)*c(x*c(x)) where c(x) is the g.f. of A000108.

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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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A372233 Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-x-x^2) )^n.

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A368931 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^3) ).

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A236342 Association types in 3-dimensional algebra.

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A300474 Number of partitions of the square resulting from a sequence of n n-sections, each of which divides any part perpendicular to any of the axes.

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A368932 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^4) ).

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A368933 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^5) ).

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A129442 Expansion of c(x)c(xc(x)) where c(x) is the g.f. of A000108.