A256061 -id:A256061 - 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

A255982 Number T(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, such that each axis is used at least once; 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, 29, 30, 0, 14, 184, 486, 336, 0, 42, 1148, 5880, 9744, 5040, 0, 132, 7228, 64464, 192984, 230400, 95040, 0, 429, 46224, 679195, 3279060, 6792750, 6308280, 2162160, 0, 1430, 300476, 7043814, 51622600, 165293700, 259518600, 196756560, 57657600

Offset: 0

Alois P. Heinz, Mar 13 2015

			A(3,1) = 5:
  [||-|---], [-|||---], [-|-|-|-], [---|||-], [---|-||].
.
A(2,2) = 4:
  ._______.  ._______.  ._______.  ._______.
  |   |   |  |   |   |  |   |   |  |       |
  |___|   |  |   |___|  |___|___|  |_______|
  |   |   |  |   |   |  |       |  |   |   |
  |___|___|  |___|___|  |_______|  |___|___|.
.
Triangle T(n,k) begins:
  1
  0,   1;
  0,   2,     4;
  0,   5,    29,     30;
  0,  14,   184,    486,     336;
  0,  42,  1148,   5880,    9744,    5040;
  0, 132,  7228,  64464,  192984,  230400,   95040;
  0, 429, 46224, 679195, 3279060, 6792750, 6308280, 2162160;
  ...

Alois P. Heinz, Rows n = 0..135, flattened

Columns k=0-10 give: A000007, A000108 (for n>0), A258416, A258417, A258418, A258419, A258420, A258421, A258422, A258423, A258424.
Main diagonal gives A001761.
Row sums give A258425.
T(2n,n) give A258426.
Cf. A237018, A256061, A258427.

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:= (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);

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}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 20 2016, after Alois P. Heinz *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A237018(n,k-i).

A253180 Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types that are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 5, 15, 5, 0, 14, 98, 84, 14, 0, 42, 630, 1050, 420, 42, 0, 132, 4092, 11880, 8580, 1980, 132, 0, 429, 27027, 129129, 150150, 60060, 9009, 429, 0, 1430, 181610, 1381380, 2432430, 1501500, 380380, 40040, 1430, 0, 4862, 1239810, 14707550, 37777740, 33795762, 12864852, 2246244, 175032, 4862

Offset: 0

Alois P. Heinz, Mar 23 2015

In general, column k>0 is asymptotic to (4*k)^n / (k!*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

			T(3,1) = 5: ()()(), ()(()), (())(), (()()), ((())).
T(3,2) = 15: ()()[], ()[](), ()[][], ()([]), ()[()], ()[[]], (())[], ([])(), ([])[], (()[]), ([]()), ([][]), (([])), ([()]), ([[]]).
T(3,3) = 5: ()[]{}, ()[{}], ([]){}, ([]{}), ([{}]).
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,     2;
  0,   5,    15,      5;
  0,  14,    98,     84,     14;
  0,  42,   630,   1050,    420,    42;
  0, 132,  4092,  11880,   8580,  1980,  132;
  0, 429, 27027, 129129, 150150, 60060, 9009, 429;
  ...

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

Columns k=0-10 give: A000007, A000108 (for n>0), A258390, A258391, A258392, A258393, A258394, A258395, A258396, A258397, A258398.
Main diagonal gives A000108.
First lower diagonal gives A002740(n+2).
T(2n,n) gives A258399.
Row sums give A064299.
Cf. A000142, A256061.

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/((k-i)!*i!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

A[n_, k_] := A[n, k] = k^n*CatalanNumber[n]; T[0, 0] = 1; T[n_, k_] := Sum[A[n, k-i]*(-1)^i/((k-i)!*i!), {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 11 2017, adapted from Maple *)

T(n,k) = A256061(n,k)/k! = Sum_{i=0..k} (-1)^i * C(k,i) * (k-i)^n * A000108(n) / A000142(n).

A290605 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 2/(1 + sqrt(1 - 4kx)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 8, 5, 0, 1, 4, 18, 40, 14, 0, 1, 5, 32, 135, 224, 42, 0, 1, 6, 50, 320, 1134, 1344, 132, 0, 1, 7, 72, 625, 3584, 10206, 8448, 429, 0, 1, 8, 98, 1080, 8750, 43008, 96228, 54912, 1430, 0, 1, 9, 128, 1715, 18144, 131250, 540672, 938223, 366080, 4862, 0

Offset: 0

Ilya Gutkovskiy, Aug 07 2017

Number of 2n-length strings of balanced parentheses of at most k different types. Also number of binary trees with n inner nodes of at most k different dimensions. - Alois P. Heinz, Oct 28 2019

			G.f. of column k: A(x) = 1 + k*x + 2*k^2*x^2 + 5*k^3*x^3 + 14*k^4*x^4 + 42*k^5*x^5 + 132*k^6*x^6 + ...
Square array begins:
  1,   1,     1,      1,      1,       1,  ...
  0,   1,     2,      3,      4,       5,  ...
  0,   2,     8,     18,     32,      50,  ...
  0,   5,    40,    135,    320,     625,  ...
  0,  14,   224,   1134,   3584,    8750,  ...
  0,  42,  1344,  10206,  43008,  131250,  ...

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

Columns k=0-10 give:  A000007, A000108, A151374, A005159, A151403, A156058, A156128, A156266, A156270, A156273, A156275.
Rows n=0-2 give: A000012, A001477, A001105.
Main diagonal gives A291699.
Cf. A000108, A253180, A256061.

ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Oct 28 2019

Table[Function[k, SeriesCoefficient[2/(1 + Sqrt[1 - 4 k x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

A(n,k) = k^n*(2*n)!/(n!*(n + 1)!).
A(n,k) = k^n*A000108(n).
G.f. of column k: 2/(1 + sqrt(1 - 4*k*x)).
G.f. of column k: 1/(1 - k*x/(1 - k*x/(1 - k*x/(1 - k*x/(1 - k*x/(1 - ...)))))), a continued fraction.
E.g.f. of column k: (BesselI(0,2*k*x) - BesselI(1,2*k*x))*exp(2*k*x).
If g.f. = 2/(1 + sqrt(1 - 4*k*x)), then a(n) ~ k^n*4^n/(sqrt(Pi)*n^(3/2)).
A(n,k) = Sum_{i=0..k} binomial(k,i) * A256061(n,k-i). - Alois P. Heinz, Oct 28 2019
For fixed k >= 1, Sum_{n>=0} 1/A(n,k) = 2*k*(8*k + 1) / (4*k - 1)^2 + 24 * k^2 * arcsin(1/(2*sqrt(k))) / (4*k - 1)^(5/2). - Vaclav Kotesovec, Nov 23 2021
For fixed k >= 1, Sum_{n>=0} (-1)^n / A(n,k) = 2*k*(8*k - 1) / (4*k + 1)^2 - 24 * k^2 * log((1 + sqrt(4*k + 1))/(2*sqrt(k))) / (4*k + 1)^(5/2). - Vaclav Kotesovec, Nov 24 2021

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).

cp's OEIS Frontend

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

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A255982 Number T(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, such that each axis is used at least once; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A253180 Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types that are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A290605 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 2/(1 + sqrt(1 - 4*k*x)).

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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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A290605 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 2/(1 + sqrt(1 - 4kx)).