%I #116 Jun 13 2024 11:38:31
%S 1,1,4,25,196,1764,17424,184041,2044900,23639044,282105616,3455793796,
%T 43268992144,551900410000,7152629313600,93990019574025,
%U 1250164827828900,16807771574144100,228138727737690000,3123219182728976100,43087676888260976400,598598221893939680400,8369059450146650049600
%N Squares of Catalan numbers.
%C Also multi-component meanders.
%C 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
%C 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
%C 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
%H T. D. Noe, <a href="/A001246/b001246.txt">Table of n, a(n) for n = 0..100</a>
%H Andrei Asinowski, Cyril Banderier, and Sarah J. Selkirk, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2023/30.pdf">From Kreweras to Gessel: A walk through patterns in the quarter plane</a>, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #30.
%H Cyril Banderier, Markus Kuba, Stephan Wagner, and Michael Wallner, <a href="https://arxiv.org/abs/2311.17226">Composition schemes: q-enumerations and phase transitions</a>, arXiv:2311.17226 [math.CO], 2023. See p. 6.
%H Mireille Bousquet-Mélou and Marni Mishna, <a href="http://arxiv.org/abs/0810.4387">Walks with small steps in the quarter plane</a>, arXiv:0810.4387 [math.CO], 2008-2009.
%H P. Di Francesco, O. Golinelli, and E. Guitter, <a href="http://arXiv.org/abs/hep-th/9506030">Meander, folding and arch statistics</a>, arXiv:hep-th/9506030, 1995.
%H David E. Evans and Mathew Pugh, <a href="http://arxiv.org/abs/1404.1877">Spectral measures associated to rank two Lie groups and finite subgroups of GL(2,Z)</a>, arXiv preprint arXiv:1404.1877 [math.OA], 2014-2015.
%H O. Guibert, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00121-1">Stack words, standard Young tableaux, permutations with forbidden subsequences and planar maps</a>, Discr. Math., Vol. 210, No. 1-3 (2000), pp. 71-85.
%H Hung Phuc Hoang and Torsten Mütze, <a href="https://arxiv.org/abs/1911.12078">Combinatorial generation via permutation languages. II. Lattice congruences</a>, arXiv:1911.12078 [math.CO], 2019.
%H Amya Luo, <a href="https://math.dartmouth.edu/theses/undergrad/2024/Luo-thesis.pdf">Pattern Avoidance in Nonnesting Permutations</a>, Undergraduate Thesis, Dartmouth College (2024). See p. 11.
%H Tilman Piesk, <a href="http://en.wikiversity.org/wiki/Partition_related_number_triangles#rot">Partition related number triangles</a>. (Wikiversity article)
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Sato_series">Ramanujan-Sato series</a>.
%F 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
%F 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
%F 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
%F D-finite with recurrence (n+1)^2*a(n) -4*(2*n-1)^2*a(n-1)=0. - _R. J. Mathar_, Jan 04 2013
%F From _Ilya Gutkovskiy_, Mar 23 2017: (Start)
%F a(n) ~ 16^n/(Pi*n^3).
%F Sum_{n>=0} 1/a(n) = 3F2(1,2,2; 1/2,1/2; 1/16) = 2.295732295098655... (End)
%F Sum {n>=0} a(n)*(n+1)/16^n = 4/Pi. This is a kind of Ramanujan-Sato series. - _Ralf Steiner_, Mar 23 2017
%F From _Peter Bala_, Mar 28 2018: (Start)
%F 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!).
%F a(n) = 1/(2*n + 1)*A000891(n).
%F a(n) = (n + 2)/(2*n + 1)*A000356(n).
%F a(n) = (n + 2)/3*A186264(n-1). (End)
%F From _Amiram Eldar_, Mar 27 2022: (Start)
%F a(n) = A000108(n)^2.
%F Sum_{n>=0} a(n)/16^n = 16/Pi - 4. (End)
%p seq((binomial(2*n,n)/(1+n))^2, n=0..18); # _Zerinvary Lajos_, Jun 18 2007
%t 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 *)
%t CatalanNumber[Range[0,30]]^2 (* _Harvey P. Dale_, Apr 26 2011 *)
%t a[ n_] := If[ n == -1, 0, CatalanNumber[ n]^2] (* _Michael Somos_, Jul 11 2011 *)
%t 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 *)
%t 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 *)
%o (MuPAD) combinat::dyckWords::count(n)^2 $ n = 0..18 // _Zerinvary Lajos_, Feb 15 2007
%o (Sage) [catalan_number(i)^2 for i in range(0,19)] # _Zerinvary Lajos_, May 17 2009
%o (PARI) a(n)=(binomial(2*n,n)/(n+1))^2 \\ _Charles R Greathouse IV_, Jul 16 2011
%o (GAP) List([0..25],n->(Binomial(2*n,n)/(n+1))^2); # _Muniru A Asiru_, Mar 28 2018
%Y Cf. A000108, A000356, A000891, A186264.
%Y Row sums of triangle A008828.
%Y Probably diagonal of A209805.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E 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