%I M2998 N1214 #298 Apr 05 2024 11:10:14

%S 1,3,15,93,639,4653,35169,272835,2157759,17319837,140668065,

%T 1153462995,9533639025,79326566595,663835030335,5582724468093,

%U 47152425626559,399769750195965,3400775573443089,29016970072920387,248256043372999089

%N a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(2*k,k).

%C This is the Taylor expansion of a special point on a curve described by Beauville. - _Matthijs Coster_, Apr 28 2004

%C a(n) is the 2n-th moment of the distance from the origin of a 3-step random walk in the plane. - Peter M. W. Gill (peter.gill(AT)nott.ac.uk), Feb 27 2004

%C a(n) is the number of Abelian squares of length 2n over a 3-letter alphabet. - _Jeffrey Shallit_, Aug 17 2010

%C Consider 2D simple random walk on honeycomb lattice. a(n) gives number of paths of length 2n ending at origin. - _Sergey Perepechko_, Feb 16 2011

%C Row sums of A318397 the square of A008459. - _Peter Bala_, Mar 05 2013

%C Conjecture: For each n=1,2,3,... the polynomial g_n(x) = Sum_{k=0..n} binomial(n,k)^2*binomial(2k,k)*x^k is irreducible over the field of rational numbers. - _Zhi-Wei Sun_, Mar 21 2013

%C This is one of the Apery-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017

%C a(n) is the sum of the squares of the coefficients of (x + y + z)^n. - _Michael Somos_, Aug 25 2018

%C a(n) is the constant term in the expansion of (1 + (1 + x) * (1 + y) + (1 + 1/x) * (1 + 1/y))^n. - _Seiichi Manyama_, Oct 28 2019

%D Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Seiichi Manyama, <a href="/A002893/b002893.txt">Table of n, a(n) for n = 0..1051</a> (terms 0..100 from T. D. Noe)

%H B. Adamczewski, J. P. Bell, and E. Delaygue, <a href="https://arxiv.org/abs/1603.04187">Algebraic independence of G-functions and congruences à la Lucas"</a>, arXiv preprint arXiv:1603.04187 [math.NT], 2016.

%H David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, <a href="http://arxiv.org/abs/0801.0891">Elliptic integral evaluations of Bessel moments</a>, arXiv:0801.0891 [hep-th], 2008.

%H P. Barrucand, <a href="http://dx.doi.org/10.1137/1017013">A combinatorial identity, Problem 75-4</a>, SIAM Rev., 17 (1975), 168. <a href="http://dx.doi.org/10.1137/1018056">Solution</a> by D. R. Breach, D. McCarthy, D. Monk and P. E. O'Neil, SIAM Rev. 18 (1976), 303.

%H P. Barrucand, <a href="/A002893/a002893.pdf">Problem 75-4, A Combinatorial Identity</a>, SIAM Rev., 17 (1975), 168. [Annotated scanned copy of statement of problem]

%H Arnaud Beauville, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5543443c/f31.item">Les familles stables de courbes sur P_1 admettant quatre fibres singulières</a>, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982.

%H Artur Bille, Victor Buchstaber, Simon Coste, Satoshi Kuriki, and Evgeny Spodarev, <a href="https://arxiv.org/abs/2306.01462">Random eigenvalues of graphenes and the triangulation of plane</a>, arXiv:2306.01462 [math.SP], 2023.

%H Jonathan M. Borwein, <a href="https://carmamaths.org/resources/jon/beauty.pdf">A short walk can be beautiful</a>, 2015.

%H Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, <a href="http://www.carmamaths.org/resources/jon/walks.pdf">Random Walk Integrals</a>, 2010.

%H Jonathan M. Borwein and Armin Straub, <a href="http://carmamaths.org/resources/jon/wmi-paper.pdf">Mahler measures, short walks and log-sine integrals</a>.

%H Jonathan M. Borwein, Armin Straub and Christophe Vignat, <a href="http://carmamaths.org/resources/jon/dwalks.pdf">Densities of short uniform random walks, Part II: Higher dimensions</a>, Preprint, 2015

%H Jonathan M. Borwein, Armin Straub and James Wan, <a href="http://dx.doi.org/10.1080/10586458.2013.748379">Three-Step and Four-Step Random Walk Integrals</a>, Exper. Math., 22 (2013), 1-14.

%H Charles Burnette and Chung Wong, <a href="https://arxiv.org/abs/1609.05580">Abelian Squares and Their Progenies</a>, arXiv:1609.05580 [math.CO], 2016.

%H David Callan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Callan2/callan204.html">A combinatorial interpretation for an identity of Barrucand</a>, JIS 11 (2008) 08.3.4.

%H Shaun Cooper, <a href="https://arxiv.org/abs/2302.00757">Apéry-like sequences defined by four-term recurrence relations</a>, arXiv:2302.00757 [math.NT], 2023.

%H M. Coster, <a href="/A001850/a001850_1.pdf">Email, Nov 1990</a>

%H Eric Delaygue, <a href="http://arxiv.org/abs/1310.4131">Arithmetic properties of Apery-like numbers</a>, arXiv preprint arXiv:1310.4131 [math.NT], 2013.

%H C. Domb, <a href="http://dx.doi.org/10.1080/00018736000101199">On the theory of cooperative phenomena in crystals</a>, Advances in Phys., 9 (1960), 149-361.

%H Jeffrey S. Geronimo, Hugo J. Woerdeman, and Chung Y. Wong, <a href="https://arxiv.org/abs/2101.00525">The autoregressive filter problem for multivariable degree one symmetric polynomials</a>, arXiv:2101.00525 [math.CA], 2021.

%H Ofir Gorodetsky, <a href="https://arxiv.org/abs/2102.11839">New representations for all sporadic Apéry-like sequences, with applications to congruences</a>, arXiv:2102.11839 [math.NT], 2021. See C p. 2.

%H Victor J. W. Guo, <a href="http://arxiv.org/abs/1201.0617">Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers</a>, arXiv preprint arXiv:1201.0617 [math.NT], 2012.

%H Victor J. W. Guo, Guo-Shuai Mao and Hao Pan, <a href="http://arxiv.org/abs/1511.04005">Proof of a conjecture involving Sun polynomials</a>, arXiv preprint arXiv:1511.04005 [math.NT], 2015.

%H E. Hallouin and M. Perret, <a href="http://arxiv.org/abs/1503.06591">A Graph Aided Strategy to Produce Good Recursive Towers over Finite Fields</a>, arXiv preprint arXiv:1503.06591 [math.NT], 2015.

%H J. A. Hendrickson, Jr., <a href="http://dx.doi.org/10.1080/00949659508811639">On the enumeration of rectangular (0,1)-matrices</a>, Journal of Statistical Computation and Simulation, 51 (1995), 291-313.

%H S. Herfurtner, <a href="https://doi.org/10.1007/BF01445211">Elliptic surfaces with four singular fibres</a>, Mathematische Annalen, 1991. <a href="https://archive.mpim-bonn.mpg.de/id/eprint/860/">Preprint</a>.

%H Pakawut Jiradilok and Elchanan Mossel, <a href="https://arxiv.org/abs/2402.11990">Gaussian Broadcast on Grids</a>, arXiv:2402.11990 [cs.IT], 2024. See p. 27.

%H Tanya Khovanova and Konstantin Knop, <a href="http://arxiv.org/abs/1409.0250">Coins of three different weights</a>, arXiv:1409.0250 [math.HO], 2014.

%H Murray S. Klamkin, ed., <a href="http://dx.doi.org/10.1137/1.9781611971729">Problems in Applied Mathematics: Selections from SIAM Review</a>, SIAM, 1990; see pp. 148-149.

%H Bradley Klee, <a href="/A006077/a006077.pdf">Checking Weierstrass data</a>, 2023.

%H Amita Malik and Armin Straub, <a href="https://doi.org/10.1007/s40993-016-0036-8">Divisibility properties of sporadic Apéry-like numbers</a>, Research in Number Theory, 2016, 2:5

%H Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/2006632/sum-involving-the-product-of-binomial-coefficients">sum involving the product of binomial coefficients</a>, Nov 10 2016.

%H L. B. Richmond and Jeffrey Shallit, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i1r72">Counting abelian squares</a>, Electronic J. Combinatorics 16 (1), #R72, June 2009. [From _Jeffrey Shallit_, Aug 17 2010]

%H Armin Straub, <a href="http://arminstraub.com/pub/dissertation">Arithmetic aspects of random walks and methods in definite integration</a>, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012. - From _N. J. A. Sloane_, Dec 16 2012

%H Zhi-Hong Sun, <a href="https://arxiv.org/abs/1803.10051">Congruences for Apéry-like numbers</a>, arXiv:1803.10051 [math.NT], 2018.

%H Zhi-Hong Sun, <a href="https://arxiv.org/abs/2004.07172">New congruences involving Apéry-like numbers</a>, arXiv:2004.07172 [math.NT], 2020.

%H Zhi-Wei Sun, <a href="http://math.nju.edu.cn/~zwsun/150f.pdf">Connections between p = x^2+3y^2 and Franel numbers</a>, J. Number Theory 133(2013), 2919-2928.

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1007/s11139-015-9727-3">Congruences involving g_n(x)=sum_{k=0..n}binom(n,k)^2*binom(2k,k)*x^k</a>, Ramanujan J., in press. Doi: 10.1007/s11139-015-9727-3.

%H Brani Vidakovic, <a href="https://www.jstor.org/stable/2684723">All roads lead to Rome--even in the honeycomb world</a>, Amer. Statist., 48 (1994) no. 3, 234-236.

%H Yi Wang and Bao-Xuan Zhu, <a href="http://arxiv.org/abs/1303.5595">Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences</a>, arXiv preprint arXiv:1303.5595 [math.CO], 2013.

%H D. Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/tex/AperylikeRecEqs/fulltext.pdf">Integral solutions of Apery-like recurrence equations</a>. See line C in sporadic solutions table of page 5.

%F a(n) = Sum_{m=0..n} binomial(n, m) * A000172(m). [Barrucand]

%F D-finite with recurrence: (n+1)^2 a(n+1) = (10*n^2+10*n+3) * a(n) - 9*n^2 * a(n-1). - _Matthijs Coster_, Apr 28 2004

%F Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0, 2*sqrt(x))^3. - _Vladeta Jovovic_, Mar 11 2003

%F a(n) = Sum_{p+q+r=n} (n!/(p!*q!*r!))^2 with p, q, r >= 0. - _Michael Somos_, Jul 25 2007

%F a(n) = 3*A087457(n) for n>0. - _Philippe Deléham_, Sep 14 2008

%F a(n) = hypergeom([1/2, -n, -n], [1, 1], 4). - _Mark van Hoeij_, Jun 02 2010

%F G.f.: 2*sqrt(2)/Pi/sqrt(1-6*z-3*z^2+sqrt((1-z)^3*(1-9*z))) * EllipticK(8*z^(3/2)/(1-6*z-3*z^2+sqrt((1-z)^3*(1-9*z)))). - _Sergey Perepechko_, Feb 16 2011

%F G.f.: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)*(1-x)^n / (1-3*x)^(3*n+1). - _Paul D. Hanna_, Feb 26 2012

%F Asymptotic: a(n) ~ 3^(2*n+3/2)/(4*Pi*n). - _Vaclav Kotesovec_, Sep 11 2012

%F G.f.: 1/(1-3*x)*(1-6*x^2*(1-x)/(Q(0)+6*x^2*(1-x))), where Q(k) = (54*x^3 - 54*x^2 + 9*x -1)*k^2 + (81*x^3 - 81*x^2 + 18*x -2)*k + 33*x^3 - 33*x^2 +9*x - 1 - 3*x^2*(1-x)*(1-3*x)^3*(k+1)^2*(3*k+4)*(3*k+5)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jul 16 2013

%F G.f.: G(0)/(2*(1-9*x)^(2/3)), where G(k) = 1 + 1/(1 - 3*(3*k+1)^2*x*(1-x)^2/(3*(3*k+1)^2*x*(1-x)^2 - (k+1)^2*(1-9*x)^2/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 31 2013

%F a(n) = [x^(2n)] 1/agm(sqrt((1-3*x)*(1+x)^3), sqrt((1+3*x)*(1-x)^3)). - _Gheorghe Coserea_, Aug 17 2016

%F 0 = +a(n)*(+a(n+1)*(+729*a(n+2) -1539*a(n+3) +243*a(n+4)) +a(n+2)*(-567*a(n+2) +1665*a(n+3) -297*a(n+4)) +a(n+3)*(-117*a(n+3) +27*a(n+4))) +a(n+1)*(+a(n+1)*(-324*a(n+2) +720*a(n+3) -117*a(n+4)) +a(n+2)*(+315*a(n+2) -1000*a(n+3) +185*a(n+4)) +a(n+3)*(+80*a(n+3) -19*a(n+4))) +a(n+2)*(+a(n+2)*(-9*a(n+2) +35*a(n+3) -7*a(n+4)) +a(n+3)*(-4*a(n+3) +a(n+4))) for all n in Z. - _Michael Somos_, Oct 30 2017

%F G.f. y=A(x) satisfies: 0 = x*(x - 1)*(9*x - 1)*y'' + (27*x^2 - 20*x + 1)*y' + 3*(3*x - 1)*y. - _Gheorghe Coserea_, Jul 01 2018

%F Sum_{k>=0} binomial(2*k,k) * a(k) / 6^(2*k) = A086231 = (sqrt(3)-1) * (Gamma(1/24) * Gamma(11/24))^2 / (32*Pi^3). - _Vaclav Kotesovec_, Apr 23 2023

%F From _Bradley Klee_, Jun 05 2023: (Start)

%F The g.f. T(x) obeys a period-annihilating ODE:

%F 0=3*(-1 + 3*x)*T(x) + (1 - 20*x + 27*x^2)*T'(x) + x*(-1 + x)*(-1 + 9*x)*T''(x).

%F The periods ODE can be derived from the following Weierstrass data:

%F g2 = (3/64)*(1 + 3*x)*(1 - 15*x + 75*x^2 + 3*x^3);

%F g3 = -(1/512)*(-1 + 6*x + 3*x^2)*(1 - 12*x + 30*x^2 - 540*x^3 + 9*x^4);

%F which determine an elliptic surface with four singular fibers. (End)

%e G.f.: A(x) = 1 + 3*x + 15*x^2 + 93*x^3 + 639*x^4 + 4653*x^5 + 35169*x^6 + ...

%e G.f.: A(x) = 1/(1-3*x) + 6*x^2*(1-x)/(1-3*x)^4 + 90*x^4*(1-x)^2/(1-3*x)^7 + 1680*x^6*(1-x)^3/(1-3*x)^10 + 34650*x^8*(1-x)^4/(1-3*x)^13 + ... - _Paul D. Hanna_, Feb 26 2012

%p series(1/GaussAGM(sqrt((1-3*x)*(1+x)^3), sqrt((1+3*x)*(1-x)^3)), x=0, 42) # _Gheorghe Coserea_, Aug 17 2016

%p A002893 := n -> hypergeom([1/2, -n, -n], [1, 1], 4):

%p seq(simplify(A002893(n)), n=0..20); # _Peter Luschny_, May 23 2017

%t Table[Sum[Binomial[n,k]^2 Binomial[2k,k],{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Aug 19 2011 *)

%t a[ n_] := If[ n < 0, 0, HypergeometricPFQ[ {1/2, -n, -n}, {1, 1}, 4]]; (* _Michael Somos_, Oct 16 2013 *)

%t a[n_] := SeriesCoefficient[BesselI[0, 2*Sqrt[x]]^3, {x, 0, n}]*n!^2; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Dec 30 2013 *)

%t a[ n_] := If[ n < 0, 0, Block[ {x, y, z}, Expand[(x + y + z)^n] /. {t_Integer -> t^2, x -> 1, y -> 1, z -> 1}]]; (* _Michael Somos_, Aug 25 2018 *)

%o (PARI) {a(n) = if( n<0, 0, n!^2 * polcoeff( besseli(0, 2*x + O(x^(2*n+1)))^3, 2*n))};

%o (PARI) {a(n) = sum(k=0, n, binomial(n, k)^2 * binomial(2*k, k))}; /* _Michael Somos_, Jul 25 2007 */

%o (PARI) {a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3 * x^(2*m)*(1-x)^m / (1-3*x+x*O(x^n))^(3*m+1)),n)} \\ _Paul D. Hanna_, Feb 26 2012

%o (PARI) N = 42; x='x + O('x^N); v = Vec(1/agm(sqrt((1-3*x)*(1+x)^3), sqrt((1+3*x)*(1-x)^3))); vector((#v+1)\2, k, v[2*k-1]) \\ _Gheorghe Coserea_, Aug 17 2016

%o (Magma) [&+[Binomial(n, k)^2 * Binomial(2*k, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 26 2018

%o (SageMath)

%o def A002893(n): return simplify(hypergeometric([1/2,-n,-n], [1,1], 4))

%o [A002893(n) for n in range(31)] # _G. C. Greubel_, Jan 21 2023

%Y Cf. A000172, A002895, A000984, A006480, A087457, A274600, A318397.

%Y Cf. A169714 and A169715. - _Peter Bala_, Mar 05 2013

%Y The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

%Y For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

%K nonn,easy,walk,nice

%O 0,2

%A _N. J. A. Sloane_