A001246 Squares of Catalan numbers.

1, 1, 4, 25, 196, 1764, 17424, 184041, 2044900, 23639044, 282105616, 3455793796, 43268992144, 551900410000, 7152629313600, 93990019574025, 1250164827828900, 16807771574144100, 228138727737690000, 3123219182728976100, 43087676888260976400, 598598221893939680400, 8369059450146650049600

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Also multi-component meanders.

Also, number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of 2 n steps taken from {(-1, -1), (-1, 1), (1, -1), (1, 1)}. [Evans and Pugh show that this is the same sequence.] - N. J. A. Sloane, Jul 04 2014

This is probably the diagonal of A209805. In this case a(n) = number of non-crossing partitions up to rotation of [2n+1] into n+1 blocks. See "Partition related number triangles" in Links section. - Tilman Piesk, Apr 09 2012

a(n) is also the number of regular cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_{n+1}. See Table 1 in the Hoang/Mütze reference in the Links section. - Torsten Muetze, Nov 28 2019

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• 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.
• Cyril Banderier, Markus Kuba, Stephan Wagner, and Michael Wallner, Composition schemes: q-enumerations and phase transitions, arXiv:2311.17226 [math.CO], 2023. See p. 6.
• Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
• P. Di Francesco, O. Golinelli, and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
• David E. Evans and Mathew Pugh, Spectral measures associated to rank two Lie groups and finite subgroups of GL(2,Z), arXiv preprint arXiv:1404.1877 [math.OA], 2014-2015.
• O. Guibert, Stack words, standard Young tableaux, permutations with forbidden subsequences and planar maps, Discr. Math., Vol. 210, No. 1-3 (2000), pp. 71-85.
• Hung Phuc Hoang and Torsten Mütze, Combinatorial generation via permutation languages. II. Lattice congruences, arXiv:1911.12078 [math.CO], 2019.
• Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 11.
• Tilman Piesk, Partition related number triangles. (Wikiversity article)
• Wikipedia, Ramanujan-Sato series.

Crossrefs

Cf. A000108, A000356, A000891, A186264.

Row sums of triangle A008828.

Probably diagonal of A209805.

Programs

GAP

List([0..25],n->(Binomial(2*n,n)/(n+1))^2); # Muniru A Asiru, Mar 28 2018

Maple

seq((binomial(2*n,n)/(1+n))^2, n=0..18); # Zerinvary Lajos, Jun 18 2007

Mathematica

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, 2 n], {n, 0, 25}] (* Manuel Kauers, Nov 18 2008 *)
CatalanNumber[Range[0,30]]^2  (* Harvey P. Dale, Apr 26 2011 *)
a[ n_] := If[ n == -1, 0, CatalanNumber[ n]^2] (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ (2 EllipticE[ 16 x] - (1 - 16 x) EllipticK[ 16 x] - Pi/2) / ( 2 Pi x), {x, 0, n}] (* Michael Somos, Jul 11 2011 *)
a[ n_] := If[ n < 0, 0, (2 n)! SeriesCoefficient[ HypergeometricPFQ[ {1/2}, {2, 2}, 4 x^2], {x, 0, 2 n}]] (* Michael Somos, Jul 11 2011 *)

MuPAD

combinat::dyckWords::count(n)^2 $ n = 0..18 // Zerinvary Lajos, Feb 15 2007

PARI

a(n)=(binomial(2*n,n)/(n+1))^2 \\ Charles R Greathouse IV, Jul 16 2011

Sage

[catalan_number(i)^2 for i in range(0,19)] # Zerinvary Lajos, May 17 2009

Formula

G.f.: -1/(4*x)+1/2*(16*x-1)/x * EllipticK(4*x^(1/2))/Pi + 1/x*EllipticE(4*x^(1/2))/Pi. - Vladeta Jovovic, Oct 12 2003

G.f.: 3F2( (1, 1/2, 1/2); (2, 2); 16x) = (-1 + 2F1( (-1/2, -1/2); (1); 16x))/(4*x) - Olivier Gérard, Feb 16 2011

E.g.f.: hypergeom([1/2], [2, 2], 4*x^2) = 2*BesselI(0, 2*x)^2-BesselI(0, 2*x)*BesselI(1, 2*x)/x-2*BesselI(1, 2*x)^2. - Vladeta Jovovic, Jun 04 2005

D-finite with recurrence (n+1)^2*a(n) -4*(2*n-1)^2*a(n-1)=0. - R. J. Mathar, Jan 04 2013

From Ilya Gutkovskiy, Mar 23 2017: (Start)

a(n) ~ 16^n/(Pi*n^3).

Sum_{n>=0} 1/a(n) = 3F2(1,2,2; 1/2,1/2; 1/16) = 2.295732295098655... (End)

Sum {n>=0} a(n)*(n+1)/16^n = 4/Pi. This is a kind of Ramanujan-Sato series. - Ralf Steiner, Mar 23 2017

From Peter Bala, Mar 28 2018: (Start)

a(n) = 1/(2*n + 1)*f(2*n)/(f(n)*f(n)), where f(n) = n!*(n+1)!. Cf. Catalan(n) = 1/(n + 1)*(2*n)!/(n!*n!).

a(n) = 1/(2*n + 1)*A000891(n).

a(n) = (n + 2)/(2*n + 1)*A000356(n).

a(n) = (n + 2)/3*A186264(n-1). (End)

From Amiram Eldar, Mar 27 2022: (Start)

a(n) = A000108(n)^2.

Sum_{n>=0} a(n)/16^n = 16/Pi - 4. (End)

Extensions

As a result of the work of Evans and Pugh, it was possible to merge A151342 with this sequence. - N. J. A. Sloane, Jul 04 2014