A179898 -id:A179898 - cp's OEIS frontend

A005700 a(n) = C(n)*C(n+2) - C(n+1)^2 where C() are the Catalan numbers A000108.

1, 1, 3, 14, 84, 594, 4719, 40898, 379236, 3711916, 37975756, 403127256, 4415203280, 49671036900, 571947380775, 6721316278650, 80419959684900, 977737404590100, 12058761323277900, 150656212896017400, 1904342169333848400, 24328661192286773400, 313839729380499376860

Offset: 0

N. J. A. Sloane

The old name was: Number of closed walks of 2n unit steps north, east, south, or west starting and ending at the origin and confined to the first octant.
Image of Catalan numbers (A000108) under "little Hankel" transform that sends [c_0, c_1, ...] to [d_0, d_1, ...] where d_n = c_n^2 - c_{n+1}*c_{n-1}.
The Niederhausen reference counts various classes of first octant paths by number of contacts with the line y=x. - David Callan, Sep 18 2007
In Sloane and Plouffe (1995) this was incorrectly described as "Dyck paths".
Also matchings avoiding a certain pattern (see J. Bloom and S. Elizalde). - N. J. A. Sloane, Jan 02 2013
From Bruce Westbury, Aug 22 2013: (Start)
a(n) is also the number of nested pairs of Dyck paths of length n starting and ending at the origin;
a(n) is also the number of 3-noncrossing perfect matchings on 2n points;
a(n) is also the number of 2-triangulations on n-gon;
a(n) is also the dimension of the invariant subspace of 2n-th tensor power of the spin representation of Spin(5);
a(n) is also the dimension of the invariant subspace of 2n-th tensor power of the defining representation of Sp(4). (End)
a(-1) = -3/2, a(-2) = -1/4 makes some formulas true for all n in Z. - Michael Somos, Oct 02 2014
a(n) is the number of uniquely sorted permutations of length 2n+1 that avoid the pattern 312. - Colin Defant, Jun 08 2019

			Example: a(2)=3 counts EWEW, EEWW, ENSW.
G.f. = 1 + x + 3*x^2 + 14*x^3 + 84*x^4 + 594*x^5 + 4719*x^6 + 40898*x^7 + ...

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
Andrei Asinowski, Cyril Banderier, and Sarah J. Selkirk, From Kreweras to Gessel: A walk through patterns in the quarter plane, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #30.
Jonathan Bloom and Sergi Elizalde, Pattern avoidance in matchings and partitions, arXiv preprint arXiv:1211.3442 [math.CO], 2012. See Table 1.
N. Bonichon, A bijection between realizers of maximal plane graphs and pairs of non-crossing Dyck paths, Discr. Math., 298 (2005), 104-114.
Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020. See Section 2.
W. Y. C. Cheng, E. Y. P. Deng, R. R. X. Du, R. P. Stanley and C. H. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 183).
C. P. Davis-Stober, A bijection between a set of lexicographic semiorders and pairs of non-crossing Dyck paths, Journal of Mathematical Psychology, 54, 471-474.
Myriam de Sainte-Catherine, Couplages et Pfaffiens en Combinatoire, Physique et Informatique, PhD Dissertation, Université Bordeaux I, 1983. (Annotated scanned copy of pages III.42-III.45)
Colin Defant, Catalan Intervals and Uniquely Sorted Permutations, arXiv:1904.02627 [math.CO], 2019.
Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, Bijections for Baxter families and related objects, J. Combin. Theory Ser. A, 118(3):993-1020, 2011. See Theorem 8.4.
E. Fusy, D. Poulalhon, and G. Schaeffer, Bijective counting of plane bipolar orientations and Schnyder words, Eur. J. Combinat. 30 (2009) 1646-1658, eq (2).
Juan B. Gil, Peter R. W. McNamara, Jordan O. Tirrell, and Michael D. Weiner, From Dyck paths to standard Young tableaux, arXiv:1708.00513 [math.CO], 2017.
D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, Springer 1986.
D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, Springer, 1986. (Annotated scanned copy)
D. Gouyou-Beauchamps. Standard Young tableaux of height 4 and 5, European J. Combin. 10 (1989) 69 - 82.
Nicholas M. Katz, A Note on Random Matrix Integrals, Moment Identities, and Catalan Numbers, preprint, 2015; Mathemtika, Volume 62, Issue 3 2016 , pp. 811-817.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Alec Mihailovs, Enumeration of walks on lattices, arXiv:math/9803128 (1998).
Heinrich Niederhausen, A Note on the Enumeration of Diffusion Walks in the First Octant by Their Number of Contacts with the Diagonal, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.3.
Maya Sankar, Further Bijections to Pattern-Avoiding Valid Hook Configurations, arXiv:1910.08895 [math.CO], 2019.
Robert Scherer, Topics in Number Theory and Combinatorics, Ph. D. Dissertation, Univ. of California Davis (2021).
Index entries for sequences related to Young tableaux.

A column of the triangle in A179898. A diagonal of the triangle in A185249.
Row sums of A193691, A193692. - Alois P. Heinz, Aug 03 2011
See A138349 for another version.
Cf. A000108, A006149, A006150, A248152.

LiE
```
p_tensor(2*n,[0,1],B2)|[0,0]
```
LiE
```
p_tensor(2*n,[1,0],C2)|[0,0]
```

[6*Factorial(2*n)*Factorial(2*n+2)/(Factorial(n)*Factorial(n+1)* Factorial(n+2)*Factorial(n+3)): n in [0..25]]; // Vincenzo Librandi, Aug 04 2011

CoefficientList[ Series[ HypergeometricPFQ[ {1, 1/2, 3/2}, {3, 4}, 16 x], {x, 0, 19}], x]
a[ n_] := If[ n < 1, Boole[n == 0], Det[ Table[ Binomial[i + 1, j - i + 2], {i, n}, {j, n}]]]; (* Michael Somos, Feb 25 2014 *) (* slight modification of David Callan formula *)
a[ n_] := 12 * 4^n * (2*n-1)!! * (2*n+1)!! / ((n+2)! * (n+3)!); (* Michael Somos, Oct 02 2014 *)

a(n)=6*binomial(2*n+2,n)*(2*n)!/(n+1)!/(n+3)! \\ Charles R Greathouse IV, Aug 04 2011

{a(n) = if( n<0, if( n<-2, 0, [-3/2, -1/4][-n]), 6 * (2*n)! * (2*n+2)! / (n! * (n+1)! * (n+2)! * (n+3)!))}; /* Michael Somos, Oct 02 2014 */

G.f.: 3F2( [ 1, 1/2, 3/2 ]; [ 3, 4 ]; 16 x ).
a(n) = 6*(2*n)!*(2*n+2)!/(n!*(n+1)!*(n+2)!*(n+3)!) (Mihailovs).
a(n) = Det[Table[binomial[i+1, j-i+2], {i, 1, n}, {j, 1, n}]]. - David Callan, Jul 20 2005
a(n) = b(n)b(n+1)/6 where b(n) is the superballot number A007054. - David Callan, Feb 01 2007
a(n) = A000108(n)*A000108(n+2) - A000108(n+1)^2. - Philippe Deléham, Apr 11 2007
G.f.: (1 + 6*x - hypergeom([-1/2,-3/2],[2],16*x))/(4*x^2). - Mark van Hoeij, Nov 02 2009
From Michael Somos, Oct 02 2014: (Start)
a(n) = 12 * 4^n * (2*n-1)!! * (2*n+1)!! / ((n+2)! * (n+3)!).
D-finite with recurrence 0 = a(n) * 4*(2*n+1)*(2*n+3) - a(n+1) * (n+3)*(n+4) for all n in Z.
0 = a(n)*(+65536*a(n+2) - 72192*a(n+3) + 10296*a(n+4)) + a(n+1)*(-1536*a(n+2) - 1632*a(n+3) - 282*a(n+4)) + a(n+2)*(+40*a(n+2) - 6*a(n+3) + a(n+4)) for all n in Z.
0 = a(n)^2*a(n+2)*(+1792*a(n+1) - 882*a(n+2)) + a(n)*a(n+1)^2*(+768*a(n+1) + 580*a(n+2)) + 7*a(n)*a(n+1)*a(n+2)^2 +a(n+1)^3*(-18*a(n+1) + 3*a(n+2)) for all n in Z. (End)
a(n) ~ 3 * 2^(4*n+3) / (Pi * n^5). - Vaclav Kotesovec, Feb 10 2015
From Peter Bala, Feb 22 2023: (Start)
a(n) = (12*(2*n - 1)/((n + 1)(n + 2)(n + 3))) * Catalan(n-1)*Catalan(n+1) for n >= 1.
a(n) = Product_{1 <= i <= j <= n-1} (i + j + 4)/(i + j).
a(n) = (1/2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 4)/(i + j - 1) for n >= 1. (End)
Sum_{n>=0} a(n)/16^n = 88 - 4096/(15*Pi). - Amiram Eldar, May 06 2023

More terms from James Sellers, Dec 24 1999
Corrected by Vladeta Jovovic, May 23 2004
Better definition from David Callan, Sep 18 2007
Definition simplified by N. J. A. Sloane, Nov 30 2020

A185249 Triangle read by rows: Table III.5 of Myriam de Sainte-Catherine's 1983 thesis.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 5, 0, 1, 0, 3, 0, 14, 0, 1, 1, 0, 14, 0, 42, 0, 1, 0, 4, 0, 84, 0, 132, 0, 1, 1, 0, 30, 0, 594, 0, 429, 0, 1, 0, 5, 0, 330, 0, 4719, 0, 1430, 0, 1, 1, 0, 55, 0, 4719, 0, 40898, 0, 4862, 0, 1, 0, 6, 0, 1001, 0, 81796, 0, 379236, 0, 16796, 0, 1, 1, 0, 91, 0, 26026, 0, 1643356, 0, 3711916, 0, 58786, 0, 1

Offset: 0

N. J. A. Sloane, Feb 15 2011

I have a photocopy of certain pages of the thesis, but unfortunately not enough to find the definition of this table. I have written to the author.
(Added later) However, Alois P. Heinz found a formula involving Catalan numbers which matches all the data and is surely correct, so the triangle is no longer a mystery.
Reading upwards along antidiagonals gives A123352.
From Petros Hadjicostas, Sep 04 2019: (Start)
Consider "Young tableaux with entries from the set {1,...,n}, strictly increasing in rows and not decreasing in columns. Note that usually the reverse convention between rows and columns is used."
de Sainte-Catherine and Viennot (1986) proved that "the number b_{n,k} of such Young tableaux having only columns with an even number of elements and bounded by height p = 2*k" is given by b_{n,k} = Product_{1 <= i <= j <= n} (2*k + i + j)/(i + j)." In Section 6 of their paper, they give an interpretation of this formula in terms of Pfaffians and perfect matchings.
It turns out that for the current array, T(n,k) = b_{k, (n-k)/2} if n-k is even, and 0 otherwise (for n >= 0 and 0 <= k <= n). It is unknown, however, what kind of interpretation Myriam de Sainte-Catherine gave to the number T(n,k) three years earlier in her 1983 Ph.D. dissertation. It may be distantly related to the numbers b_{n,k} that are found in her 1986 paper with G. Viennot.
(End)
The T(n, k) for n and k same parity are the numbers in the upper triangle of the Catalan Number Wall in "Number Walls in Combinatorics". Thus 0 = T(n-1, k+1)*T(n+1, k-1) - T(n-1, k-1)*T(n+1, k+1) + T(n, k)^2 for all n, k. - Michael Somos, Aug 15 2023

			Triangle begins:
  1
  0 1
  1 0  1
  0 2  0    1
  1 0  5    0     1
  0 3  0   14     0     1
  1 0 14    0    42     0       1
  0 4  0   84     0   132       0      1
  1 0 30    0   594     0     429      0       1
  0 5  0  330     0  4719       0   1430       0     1
  1 0 55    0  4719     0   40898      0    4862     0     1
  0 6  0 1001     0 81796       0 379236       0 16796     0 1
  1 0 91    0 26026     0 1643356      0 3711916     0 58786 0 1
  ...

Myriam de Sainte-Catherine, Couplages et Pfaffiens en Combinatoire, Physique et Informatique. Ph.D. Dissertation, Université Bordeaux I, 1983.

Alois P. Heinz, Rows n = 0..100, flattened
M. de Sainte-Catherine and G. Viennot, Enumeration of certain Young tableaux with bounded height, in: G. Labelle and P. Leroux (eds), Combinatoire énumérative, Lecture Notes in Mathematics, vol. 1234, Springer, Berlin, Heidelberg, 1986, pp. 58-67.
M. Somos, Number Walls in Combinatorics, 2000.

Diagonals give A000108, A005700, A006149, A006150, A006151, A000330, A006858, A091962.
Row sums give A186232. Nonzero diagonals give columns in A078920.
Cf. A179898.

with(LinearAlgebra):
ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
T := proc(n, k)
       if n=k then 1
     elif irem(n+k, 2)=1 then 0
     else Determinant(Matrix((n-k)/2, (i, j)-> ctln(i+j-1+k)))
       fi
     end:
seq(seq(T(n,k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 15 2011

t[n_, n_] = 1; t[n_, k_] /; Mod[n+k, 2] == 1 = 0; t[n_, k_] := Array[CatalanNumber[#1 + #2 - 1 + k]&, {(n-k)/2, (n-k)/2}] // Det; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 14 2014, after Alois P. Heinz *)

{T(n, k) = if((n-k)%2||k<0||k>n, 0, prod(i=1, k, prod(j=i, k, (n-k+i+j)/(i+j))))}; /* Michael Somos, Aug 15 2023 */

T(n,k) = Product_{1 <= i <= j <= k} (n-k + i + j)/(i + j) if n - k is even, and = 0 otherwise (for n >= 0 and 0 <= k <= n). - Petros Hadjicostas, Sep 04 2019

Typo in data corrected by Alois P. Heinz, Feb 15 2011

A005701 Number of exterior points formed by extending diagonals of n-gon in general position.

Original entry on oeis.org

3, 14, 40, 90, 175, 308, 504, 780, 1155, 1650, 2288, 3094, 4095, 5320, 6800, 8568, 10659, 13110, 15960, 19250, 23023, 27324, 32200, 37700, 43875, 50778, 58464, 66990, 76415, 86800, 98208, 110704, 124355

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

See Gouyou-Beauchamps for an interpretation in terms of closed paths in the first quadrant of the square grid.

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.
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..1000
Dominique Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, Springer, 1986.
Dominique Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, Springer, 1986. (Annotated scanned copy)
Index entries for sequences related to Young tableaux.

A diagonal of the triangle in A179898.

[(n+1)*(n+2)*(n+3)*(n+6)/12: n in [0..50]]; // Vincenzo Librandi, Jun 09 2013

CoefficientList[Series[(x - 3) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)

a(n) = (n+1)*(n+2)*(n+3)*(n+6)/12; \\ Michel Marcus, Dec 16 2017

a(n) = (n+1)*(n+2)*(n+3)*(n+6)/12.
G.f.: (x-3)/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
From Amiram Eldar, May 17 2025: (Start)
Sum_{n>=0} 1/a(n) = 137/300.
Sum_{n>=0} (-1)^n/a(n) = 32*log(2)/5 - 1247/300. (End)

A228334 Triangle read by rows: the X-transformation of the Catalan triangle A033184.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 14, 10, 1, 0, 84, 90, 21, 1, 0, 594, 825, 308, 36, 1, 0, 4719, 7865, 4004, 780, 55, 1, 0, 40898, 78078, 49686, 13650, 1650, 78, 1, 0, 379236, 804440, 606424, 214200, 37400, 3094, 105, 1, 0, 3711916, 8565960, 7379904, 3162816, 724812, 88179, 5320, 136, 1

Offset: 0

N. J. A. Sloane, Aug 26 2013

			Triangle begins:
  1;
  0,   1;
  0,   3,   1;
  0,  14,  10,   1;
  0,  84,  90,  21,   1;
  0, 594, 825, 308,  36,   1;
  ...

Fangfang Cai, Qing-Hu Hou, Yidong Sun, Arthur L.B. Yang, Combinatorial identities related to 2×2 submatrices of recursive matrices, arXiv:1808.05736 [math.CO], 2018.
Yidong Sun and Fei Ma, Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths, arXiv preprint arXiv:1305.2015 [math.CO], 2013.
Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013.
Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33

Cf. A033184, A179898, A228335, A228336, A228337.

nn = 9;
c[n_, k_] := Binomial[2n-k, n] (k+1)/(n+1);
a[0, 0] = 1;
a[n_, k_] := Table[c[n+k+i-1, 2k+j-1], {i, 1, 2}, {j, 1, 2}] // Det;
Table[a[n, k], {n, 0, nn}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 12 2018 *)

C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
aX(nn) = {for (n = 0, nn, for (k = 0, n, print1(matdet(matrix(2, 2, i, j, C(n+k+i-1, 2*k+j-1))), ", ");); print(););} \\ Michel Marcus, Feb 13 2014

More terms from Michel Marcus, Feb 13 2014

A-number for Catalan triangle changed by Michel Marcus, Feb 13 2014

cp's OEIS Frontend

A181571 Third column of triangle in A179898.

Views

Author

Keywords

Crossrefs

A005700 a(n) = C(n)*C(n+2) - C(n+1)^2 where C() are the Catalan numbers A000108.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

LiE

LiE

Magma

Mathematica

PARI

PARI

Formula

Extensions

A185249 Triangle read by rows: Table III.5 of Myriam de Sainte-Catherine's 1983 thesis.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A005701 Number of exterior points formed by extending diagonals of n-gon in general position.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A228334 Triangle read by rows: the X-transformation of the Catalan triangle A033184.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions