A005700 -id:A005700 - cp's OEIS frontend

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

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)

A179898 Triangle V(l,p) (l>=0, p=0..l) read by rows: see Formula for definition, see Comments for motivation.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 14, 0, 10, 0, 1, 14, 0, 40, 0, 15, 0, 1, 0, 84, 0, 90, 0, 21, 0, 1, 84, 0, 300, 0, 175, 0, 28, 0, 1, 0, 594, 0, 825, 0, 308, 0, 36, 0, 1, 594, 0, 2475, 0, 1925, 0, 504, 0, 45, 0, 1, 0, 4719, 0, 7865, 0, 4004, 0, 780, 0, 55, 0, 1, 4719, 0, 22022, 0, 21021, 0, 7644, 0, 1155, 0, 66, 0, 1, 0, 40898, 0, 78078, 0, 49686, 0, 13650, 0, 1650, 0, 78, 0, 1, 40898, 0, 208208, 0, 231868, 0, 107016, 0, 23100, 0, 2288, 0, 91, 0, 1, 0, 379236, 0, 804440, 0, 606424, 0, 214200, 0, 37400, 0, 3094, 0, 105, 0, 1

Offset: 0

N. J. A. Sloane, Jan 30 2011

If the triangle in A053121 is regarded as counting minimal subdiagonal paths in the first quadrant, this triangle enumerates pairs of non-crossing paths of the same type.

			Triangle begins:
  1;
  0, 1;
  1, 0, 1;
  0, 3, 0, 1;
  3, 0, 6, 0, 1;
  0, 14, 0, 10, 0, 1;
  14, 0, 40, 0, 15, 0, 1;
  0, 84, 0, 90, 0, 21, 0, 1;
  84, 0, 300, 0, 175, 0, 28, 0, 1;
  0, 594, 0, 825, 0, 308, 0, 36, 0, 1;
  594, 0, 2475, 0, 1925, 0, 504, 0, 45, 0, 1;
  0, 4719, 0, 7865, 0, 4004, 0, 780, 0, 55, 0, 1;
  4719, 0, 22022, 0, 21021, 0, 7644, 0, 1155, 0, 66, 0, 1;
  0, 40898, 0, 78078, 0, 49686, 0, 13650, 0, 1650, 0, 78, 0, 1;
  40898, 0, 208208, 0, 231868, 0, 107016, 0, 23100, 0, 2288, 0, 91, 0, 1;
  ...

D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableau de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986 (see |V_{l,p}| on page 114).

The first two columns are both A005700, the next column is A181571. The diagonals on the right give A000217, A005701. Row sums are A005817.

V:=proc(l,p)
if ((l-p) mod 2) = 1 then 0 else l!*(l+2)!*(p+3)! / (((l-p)/2)!*((l-p)/2+1)!*p!*((l+p)/2+2)!*((l+p)/2+3)!); fi;
end;
r:=n->[seq( V(n,p),p=0..n)];
for n from 0 to 15 do lprint(r(n)); od:

v[l_, p_] := If[Mod[l-p, 2] == 1, 0, l!*(l+2)!*(p+3)!/(((l-p)/2)!*((l-p)/2+1)!*p!*((l+p)/2+2)!*((l+p)/2+3)!)]; Table[v[l, p], {l, 0, 15}, {p, 0, l}] // Flatten (* Jean-François Alcover, Jan 09 2014, translated from Maple *)

V(l,p) = 0 if l and p have opposite parity, otherwise V(l,p) = l!*(l+2)!*(p+3)!/(((l-p)/2)!*((l-p)/2+1)!*p!*((l+p)/2+2)!*((l+p)/2+3)!).

A193691 Triangle T(n,k), n>=0, 1<=k<=C(n), read by rows: T(n,k) = number of elements <= k-th path in the poset of Dyck paths of semilength n ordered by inclusion.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 4, 5, 1, 2, 2, 4, 5, 2, 4, 4, 8, 10, 5, 10, 13, 14, 1, 2, 2, 4, 5, 2, 4, 4, 8, 10, 5, 10, 13, 14, 2, 4, 4, 8, 10, 4, 8, 8, 16, 20, 10, 20, 26, 28, 5, 10, 10, 20, 25, 13, 26, 34, 37, 14, 28, 37, 41, 42, 1, 2, 2, 4, 5, 2, 4, 4, 8, 10, 5, 10, 13, 14, 2, 4, 4, 8, 10, 4

Offset: 0

Alois P. Heinz, Aug 02 2011

			Dyck paths of semilength n=3 listed in lexicographic order:
.                                               /\
.                  /\      /\        /\/\      /  \
.     /\/\/\    /\/  \    /  \/\    /    \    /    \
.     101010    101100    110010    110100    111000
. k = (1)       (2)       (3)       (4)       (5)
.
We have (1) <= (1); (1),(2) <= (2); (1),(3) <= (3); (1),(2),(3),(4) <= (4); and (1),(2),(3),(4),(5) <= (5), thus row 3 = [1, 2, 2, 4, 5].
Triangle begins:
  1;
  1;
  1, 2;
  1, 2, 2, 4, 5;
  1, 2, 2, 4, 5, 2, 4, 4, 8, 10, 5, 10, 13, 14;
  1, 2, 2, 4, 5, 2, 4, 4, 8, 10, 5, 10, 13, 14, 2, 4, 4, 8, ...

Alois P. Heinz, Rows n = 0..9, flattened

Row sums give A005700.
Lengths and last elements of rows give A000108.
Cf. A193692, A193693, A193694.

d:= proc(n, l) local m; m:= nops(l);
      `if`(n=m, [l], [seq(d(n, [l[], j])[],
                     j=`if`(m=0, 1, max(m+1, l[-1]))..n)])
    end:
le:= proc(x, y) local i;
       for i to nops(x) do if x[i]>y[i] then return false fi od; true
     end:
T:= proc(n) option remember; local l;
      l:= d(n, []);
      seq(add(`if`(le(l[i], l[j]), 1, 0), i=1..j), j=1..nops(l))
    end:
seq(T(n), n=0..6);

d[n_, l_] := d[n, l] = Module[{m}, m = Length[l]; If[n == m, {l}, Flatten[#, 1]& @ Table[d[n, Append[l, j]], {j, If[m == 0, 1, Max[m+1, Last[l]]], n}]]]; le[x_, y_] := Module[{i}, For[i = 1, i <= Length[x], i++, If[x[[i]] > y[[i]], Return[False]]]; True]; T[n_] := T[n] = Module[{l}, l = d[n, {}]; Table[Sum[If[le[l[[i]], l[[j]]], 1, 0], {i, 1, j}], {j, 1, Length[l]}]]; Table[T[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Feb 01 2017, after Alois P. Heinz *)

A193692 Triangle T(n,k), n>=0, 1<=k<=C(n), read by rows: T(n,k) = number of elements >= k-th path in the poset of Dyck paths of semilength n ordered by inclusion.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 3, 2, 1, 14, 9, 10, 7, 4, 9, 6, 7, 5, 3, 4, 3, 2, 1, 42, 28, 32, 23, 14, 32, 22, 26, 19, 12, 17, 13, 9, 5, 28, 19, 22, 16, 10, 23, 16, 19, 14, 9, 13, 10, 7, 4, 14, 10, 12, 9, 6, 9, 7, 5, 3, 5, 4, 3, 2, 1, 132, 90, 104, 76, 48, 107, 75, 89, 66, 43, 62, 48, 34, 20, 104

Offset: 0

Alois P. Heinz, Aug 02 2011

			Dyck paths of semilength n=3 listed in lexicographic order:
.                                               /\
.                  /\      /\        /\/\      /  \
.     /\/\/\    /\/  \    /  \/\    /    \    /    \
.     101010    101100    110010    110100    111000
. k = (1)       (2)       (3)       (4)       (5)
.
We have (1),(2),(3),(4),(5) >= (1); (2),(4),(5) >= (2); (3),(4),(5) >= (3);
(4),(5) >= (4); and (5) >= (5), thus row 3 = [5, 3, 3, 2, 1].
Triangle begins:
1;
1;
2,   1;
5,   3,  3,  2,  1;
14,  9, 10,  7,  4,  9,  6,  7,  5,  3,  4,  3, 2, 1;
42, 28, 32, 23, 14, 32, 22, 26, 19, 12, 17, 13, 9, 5, 28, 19, 22, 16, ...

Alois P. Heinz, Rows n = 0..9, flattened

Row sums give A005700.
Lengths and first elements of rows give A000108.
Cf. A193691, A193693, A193694.

d:= proc(n, l) local m; m:= nops(l);
      `if`(n=m, [l], [seq(d(n, [l[], j])[],
                     j=`if`(m=0, 1, max(m+1, l[-1]))..n)])
    end:
le:= proc(x, y) local i;
       for i to nops(x) do if x[i]>y[i] then return false fi od; true
     end:
T:= proc(n) option remember; local l;
      l:= d(n, []);
      seq(add(`if`(le(l[j], l[i]), 1, 0), i=j..nops(l)), j=1..nops(l))
    end:
seq(T(n), n=0..6);

d[n_, l_] := d[n, l] = Module[{m}, m = Length[l]; If[n == m, {l}, Flatten[#, 1]& @ Table[d[n, Append[l, j]], {j, If[m == 0, 1, Max[m + 1, Last[l]]], n}]]]; le[x_, y_] := Module[{i}, For[i = 1, i <= Length[x], i++, If[x[[i]] > y[[i]] , Return[False]]]; True]; T[n_] := T[n] = Module[{l}, l = d[n, {}]; Table[Sum[If[le[l[[j]], l[[i]]], 1, 0], {i, j, Length[l]}], {j, 1, Length[l]}]]; Table[T[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Feb 01 2017, after Alois P. Heinz *)

A194091 The number of labeled biconnected squaregraphs with perimeter 2*n.

Original entry on oeis.org

1, 1, 3, 14, 82, 554, 4132, 33154, 281459, 2499523

Offset: 1

Don Knuth, Aug 15 2011

Conjecture: The next terms are 23031395, 218862682, 2134993642, 21301755122, 216752986922, 2243977451994, 23590001630055, 251411579942151. - Michael D. Weiner, Apr 07 2016

Conjecture: a(n) is also the number of connected 3-noncrossing perfect matchings on [2n] (compare to A005700). - Michael D. Weiner, Jun 09 2017

			[See A194090 for the definition of "labeled squaregraph".] For n=4 the a(4)=14 labeled biconnected squaregraphs of perimeter 8 are the straight tromino (with 4 labelings), the L tromino (with 8), and the square tetromino (with 2).

Don Knuth, squaregraph.w (CWEB program).
Robert Scherer, Topics in Number Theory and Combinatorics, Ph. D. Dissertation, Univ. of California Davis (2021).

Cf. A194088, A194089, A194090, A194092, A194093.

CWEB
```
(see Knuth link).
```

G.f.: A(x) satisfies A(x*T(x)^2) = T(x)-1 where T(x) is the o.g.f. for A005700 (conjectured). - Michael D. Weiner, Jun 09 2017

Conjectured terms removed from data by Michael D. Weiner, Aug 03 2017

A138349 Moment sequence of tr(A) in USp(4).

Original entry on oeis.org

1, 0, 1, 0, 3, 0, 14, 0, 84, 0, 594, 0, 4719, 0, 40898, 0, 379236, 0, 3711916, 0, 37975756, 0, 403127256, 0, 4415203280, 0, 49671036900, 0, 571947380775, 0, 6721316278650, 0, 80419959684900, 0, 977737404590100, 0, 12058761323277900, 0

Offset: 0

Andrew V. Sutherland, Mar 16 2008

An aerated version of A005700, which is the main entry for this sequence.
If A is a random matrix in the compact group USp(4) (4 X 4 complex matrices which are unitary and symplectic), then a(n)=E[(tr(A))^n] is the n-th moment of the trace of A.
The multiplicity of the trivial representation in the n-th tensor power of the standard representation of USp(4).
Number of returning NESW walks of length n on a 2-d integer lattice remaining in the chamber x>=y>=0, same as A005700(n/2) for n even.
Under a generalized Sato-Tate conjecture, this is the moment sequence of the distribution of scaled Frobenius traces a_p/sqrt(p) (as p varies), for almost all genus 2 curves. - Andrew V. Sutherland, Mar 16 2008

			a(4)=3 because E[(tr(A))^4] = 3 for a random matrix A in USp(4).
a(4)=3 because A126120(4)A126120(8)-A126120(6)^2 = 2*14-5*5 = 3.
a(4)=3 because EEWW, EWEW and ENSW are the returning walks on Z^2 with x>=y>=0.

Bostan, Alin ; Chyzak, Frédéric; van Hoeij, Mark; Kauers, Manuel; Pech, Lucien Hypergeometric expressions for generating functions of walks with small steps in the quarter plane. Eur. J. Comb. 61, 242-275 (2017), Table 3
David J. Grabiner and Peter Magyar, Random walks in Weyl chambers and the decomposition of tensor powers, Journal of Algebraic Combinatorics, vol. 2 (1993), no. 3, pp 239-260.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.

Cf. A005700, A126120, A000108.

a(n) = (1/2)Integral_{x=0..Pi,y=0..Pi}(2cos(x)+2cos(y))^n(2cos(x)-2cos(y))^2(2/Pi*sin^2(x))(2/Pi*sin^2(y))dxdy.
a(n) = A126120(n)*A126120(n+4)-A126120(n+2)^2.
a(2n) = A005700(n) = A000108(n)*A000108(n+2)-A000108(n+1)^2, a(2n+1)=0.

A248152 a(n) = 48 * 4^n * (2n-1)!! (2n+3)!! / ((n+2)! (n+4)!).

Original entry on oeis.org

3, 4, 14, 72, 462, 3432, 28314, 252824, 2401828, 23984688, 249554968, 2687515040, 29802622140, 338931781200, 3940081956450, 46695460462200, 562939717794300, 6890720756158800, 85507580292334200, 1074244300649863200, 13647785546892580200, 175166360584464768480

Offset: 0

Michael Somos, Oct 02 2014

nonn

a(-1) = -2, a(-2) = 1/2 makes some formulas true for all n in Z.

			G.f. = 3 + 4*x + 14*x^2 + 72*x^3 + 462*x^4 + 3432*x^5 + 28314*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..200

Cf. A000108, A005700.

[24*Factorial(2*n)*Factorial(2*n+3)/(Factorial(n)*Factorial(n+1)* Factorial(n+2)*Factorial(n+4)): n in [0..30]]; // G. C. Greubel, Aug 04 2018

a[ n_] := 48 4^n (2 n - 1)!! * (2 n + 3)!! / ((n + 2)! (n + 4)!);

{a(n) = if( n<0, if( n<-2, 0, [-2, 1/2][-n]), 24 * (2*n)! * (2*n+3)! / ( n! * (n+1)! * (n+2)! * (n+4)! ))};

a(n) = C(n)*C(n+3) - C(n+1)*C(n+2) for n>=0 where C() is the Catalan numbers A000108.
a(n)*a(n+1) = 4*A005700(n)*A005700(n+2) for all n in Z.
a(n) = 24 * (2*n)! * (2*n+3)! / ( n! * (n+1)! * (n+2)! * (n+4)! ) for n>=0.
(n+4)*(n+2)*a(n) -4*(2*n+3)*(2*n-1)*a(n-1)=0.
0 = a(n)*(+3840*a(n+3) - 660*a(n+4)) + a(n+1)*(+256*a(n+2) + 144*a(n+3) + 15*a(n+4)) + a(n+2)*(+4*a(n+2) + a(n+3)) for all n in Z.
0 = a(n)^2*(+196608*a(n+1)^2 - 14336*a(n+1)*a(n+2) - 25872*a(n+2)^2) + a(n)*a(n+1)*(+67584*a(n+1)^2 + 16672*a(n+1)*a(n+2) - 56*a(n+2)^2) + a(n+1)^2*(+1008*a(n+1)^2 + 264*a(n+1)*a(n+2) + 3*a(n+2)^2) for all n in Z.

A338727 a(n) = C(n+1)^2 - 2C(n+1)C(n) + C(n)^2, where C() is a Catalan number.

Original entry on oeis.org

1, 1, 9, 81, 784, 8100, 88209, 1002001, 11778624, 142420356, 1763160100, 22268399076, 286105172544, 3730846771600, 49286086364025, 658580586980625, 8890060997894400, 121099761397088100, 1663131325207760100, 23009839285003272900, 320486852887891560000, 4491184012659823424400, 63291012091734041000100

Offset: 0

N. J. A. Sloane, Nov 30 2020

nonn

Cf. A000108, A005700, A213600.

Equals A000245(n) squared for n >= 1.

A338727 := proc(n)
    if n = 0 then
        1;
    else
        (A000108(n+1)-A000108(n))^2 ;
    end if;
end proc:
seq( A338727(n),n=0..30) ; # R. J. Mathar, Dec 11 2020

D-finite with recurrence (n+2)^2*(n-1)^2*a(n) -4*n^2*(2*n-1)^2*a(n-1)=0. - R. J. Mathar, Dec 11 2020

a(n) = A213600(2n,2) for n>=1. - Alois P. Heinz, Oct 07 2022

A355281 Number of pairs of nested Dyck paths from (0,0) to (n,n) such that the upper path only touches the diagonal at its endpoints.

Original entry on oeis.org

1, 1, 2, 9, 55, 400, 3266, 28999, 274537, 2734885, 28401315, 305352146, 3380956839, 38394091370, 445702108969, 5274935433915, 63507021523471, 776347636736261, 9621502184089320, 120726786082609207, 1531938384684090884, 19639252409244653785, 254143269904958943103, 3317204158078663935592

Offset: 0

Joel B. Lewis, Jun 26 2022

nonn

Let B be the 2 X n X n box of integer points with opposite corners (0, 0, 0) and (1, n - 1, n - 1). For n >= 1, a(n) is also the number of plane partitions that fit inside B and whose cells lie on or below the plane x + y + z = n - 1. Proof: after rotating by 90 degrees, the upper Dyck path is the outer boundary of the region of the plane partition filled with 2's and the lower Dyck path is the outer boundary of the region of the plane partition filled with 1's or 2's.

Alois P. Heinz, Table of n, a(n) for n = 0..841

Cf. A000108, A005700.

Column k=2 of A378112.

b:= proc(n) option remember; `if`(n=0, 1,
      b(n-1)*((4*n)^2-4)/(n+2)/(n+3))
    end:
a:= proc(n) option remember;
      b(n)-add(a(n-i)*b(i), i=1..n-1)
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 26 2022

nmax = 23;
c = CatalanNumber;
B[x_] = Sum[(c[n] c[n+2] - c[n+1]^2) x^n, {n, 0, nmax}];
CoefficientList[2 - 1/B[x] + O[x]^(nmax+1), x] (* Jean-François Alcover, Jul 06 2022 *)

G.f.: 2 - 1/B(x) where B(x) is the generating function for A005700.

INVERTi transform of A005700.

cp's OEIS Frontend

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

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

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A179898 Triangle V(l,p) (l>=0, p=0..l) read by rows: see Formula for definition, see Comments for motivation.

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A193691 Triangle T(n,k), n>=0, 1<=k<=C(n), read by rows: T(n,k) = number of elements <= k-th path in the poset of Dyck paths of semilength n ordered by inclusion.

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A193692 Triangle T(n,k), n>=0, 1<=k<=C(n), read by rows: T(n,k) = number of elements >= k-th path in the poset of Dyck paths of semilength n ordered by inclusion.

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A194091 The number of labeled biconnected squaregraphs with perimeter 2*n.

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CWEB

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A138349 Moment sequence of tr(A) in USp(4).

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A248152 a(n) = 48 * 4^n * (2n-1)!! (2n+3)!! / ((n+2)! (n+4)!).

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