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

0,3

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.

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.

Tilman Piesk, Partition related number triangles. (Wikiversity article)

Wikipedia, Ramanujan-Sato series.

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)

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 *)

Prog

(MuPAD) combinat::dyckWords::count(n)^2 $ n = 0..18 // Zerinvary Lajos, Feb 15 2007
(Sage) [catalan_number(i)^2 for i in range(0, 19)] # Zerinvary Lajos, May 17 2009
(PARI) a(n)=(binomial(2*n, n)/(n+1))^2 \\ Charles R Greathouse IV, Jul 16 2011
(GAP) List([0..25], n->(Binomial(2*n, n)/(n+1))^2); # Muniru A Asiru, Mar 28 2018

Crossrefs

Cf. A000108, A000356, A000891, A186264.

Row sums of triangle A008828.

Probably diagonal of A209805.

Sequence in context: A051500 A206179 A151342 * A202827 A065735 A212694

Adjacent sequences: A001243 A001244 A001245 * A001247 A001248 A001249

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

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

Status

approved