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%I A002893 M2998 N1214 #303 Nov 01 2024 02:54:33
%S A002893 1,3,15,93,639,4653,35169,272835,2157759,17319837,140668065,
%T A002893 1153462995,9533639025,79326566595,663835030335,5582724468093,
%U A002893 47152425626559,399769750195965,3400775573443089,29016970072920387,248256043372999089
%N A002893 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(2*k,k).
%C A002893 This is the Taylor expansion of a special point on a curve described by Beauville. - _Matthijs Coster_, Apr 28 2004
%C A002893 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 A002893 a(n) is the number of Abelian squares of length 2n over a 3-letter alphabet. - _Jeffrey Shallit_, Aug 17 2010
%C A002893 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 A002893 Row sums of A318397 the square of A008459. - _Peter Bala_, Mar 05 2013
%C A002893 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 A002893 This is one of the Apery-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017
%C A002893 a(n) is the sum of the squares of the coefficients of (x + y + z)^n. - _Michael Somos_, Aug 25 2018
%C A002893 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%D A002893 Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
%D A002893 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002893 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 Frits Beukers and Jan Stienstra, <a href="https://eudml.org/doc/163967">On the Picard-Fuchs Equation and the Formal Brauer Group of Certain Elliptic K3-Surfaces</a>, Mathematische Annalen (1985), Vol. 271, pp. 269-304 (see Part III).
%H A002893 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 A002893 Jonathan M. Borwein, <a href="https://carmamaths.org/resources/jon/beauty.pdf">A short walk can be beautiful</a>, 2015.
%H A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 M. Coster, <a href="/A001850/a001850_1.pdf">Email, Nov 1990</a>
%H A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 Bradley Klee, <a href="/A006077/a006077.pdf">Checking Weierstrass data</a>, 2023.
%H A002893 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 A002893 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 A002893 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 A002893 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 A002893 Zhi-Hong Sun, <a href="https://arxiv.org/abs/1803.10051">Congruences for Apéry-like numbers</a>, arXiv:1803.10051 [math.NT], 2018.
%H A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 a(n) = Sum_{m=0..n} binomial(n, m) * A000172(m). [Barrucand]
%F A002893 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 A002893 Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0, 2*sqrt(x))^3. - _Vladeta Jovovic_, Mar 11 2003
%F A002893 a(n) = Sum_{p+q+r=n} (n!/(p!*q!*r!))^2 with p, q, r >= 0. - _Michael Somos_, Jul 25 2007
%F A002893 a(n) = 3*A087457(n) for n>0. - _Philippe Deléham_, Sep 14 2008
%F A002893 a(n) = hypergeom([1/2, -n, -n], [1, 1], 4). - _Mark van Hoeij_, Jun 02 2010
%F A002893 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 A002893 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 A002893 Asymptotic: a(n) ~ 3^(2*n+3/2)/(4*Pi*n). - _Vaclav Kotesovec_, Sep 11 2012
%F A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 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 A002893 From _Bradley Klee_, Jun 05 2023: (Start)
%F A002893 The g.f. T(x) obeys a period-annihilating ODE:
%F A002893 0=3*(-1 + 3*x)*T(x) + (1 - 20*x + 27*x^2)*T'(x) + x*(-1 + x)*(-1 + 9*x)*T''(x).
%F A002893 The periods ODE can be derived from the following Weierstrass data:
%F A002893 g2 = (3/64)*(1 + 3*x)*(1 - 15*x + 75*x^2 + 3*x^3);
%F A002893 g3 = -(1/512)*(-1 + 6*x + 3*x^2)*(1 - 12*x + 30*x^2 - 540*x^3 + 9*x^4);
%F A002893 which determine an elliptic surface with four singular fibers. (End)
%e A002893 G.f.: A(x) = 1 + 3*x + 15*x^2 + 93*x^3 + 639*x^4 + 4653*x^5 + 35169*x^6 + ...
%e A002893 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 A002893 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 A002893 := n -> hypergeom([1/2, -n, -n], [1, 1], 4):
%p A002893 seq(simplify(A002893(n)), n=0..20); # _Peter Luschny_, May 23 2017
%t A002893 Table[Sum[Binomial[n,k]^2 Binomial[2k,k],{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Aug 19 2011 *)
%t A002893 a[ n_] := If[ n < 0, 0, HypergeometricPFQ[ {1/2, -n, -n}, {1, 1}, 4]]; (* _Michael Somos_, Oct 16 2013 *)
%t A002893 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 A002893 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 A002893 (PARI) {a(n) = if( n<0, 0, n!^2 * polcoeff( besseli(0, 2*x + O(x^(2*n+1)))^3, 2*n))};
%o A002893 (PARI) {a(n) = sum(k=0, n, binomial(n, k)^2 * binomial(2*k, k))}; /* _Michael Somos_, Jul 25 2007 */
%o A002893 (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 A002893 (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 A002893 (Magma) [&+[Binomial(n, k)^2 * Binomial(2*k, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 26 2018
%o A002893 (SageMath)
%o A002893 def A002893(n): return simplify(hypergeometric([1/2,-n,-n], [1,1], 4))
%o A002893 [A002893(n) for n in range(31)] # _G. C. Greubel_, Jan 21 2023
%Y A002893 Cf. A000172, A002895, A000984, A006480, A087457, A274600, A318397.
%Y A002893 Cf. A169714 and A169715. - _Peter Bala_, Mar 05 2013
%Y A002893 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 A002893 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 A002893 nonn,easy,walk,nice
%O A002893 0,2
%A A002893 _N. J. A. Sloane_

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