cp's OEIS Frontend

A002895 Domb numbers: number of 2n-step polygons on diamond lattice.

%I A002895 M3626 N1473 #249 Jun 23 2025 14:49:00
%S A002895 1,4,28,256,2716,31504,387136,4951552,65218204,878536624,12046924528,
%T A002895 167595457792,2359613230144,33557651538688,481365424895488,
%U A002895 6956365106016256,101181938814289564,1480129751586116848,21761706991570726096,321401321741959062016
%N A002895 Domb numbers: number of 2n-step polygons on diamond lattice.
%C A002895 a(n) is the (2n)th moment of the distance from the origin of a 4-step random walk in the plane. - Peter M.W. Gill (peter.gill(AT)nott.ac.uk), Mar 03 2004
%C A002895 Row sums of the cube of A008459. - _Peter Bala_, Mar 05 2013
%C A002895 Conjecture: Let D(n) be the (n+1) X (n+1) Hankel-type determinant with (i,j)-entry equal to a(i+j) for all i,j = 0..n. Then the number D(n)/12^n is always a positive odd integer. - _Zhi-Wei Sun_, Aug 14 2013
%C A002895 It appears that the expansions exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 4*x + 22*x^2 + 152*x^3 + 1241*x^4 + ... and exp( Sum_{n >= 1} 1/4*a(n)*x^n/n ) = 1 + x + 4*x^2 + 25*x^3 + 199*x^4 + ... have integer coefficients. See A267219. - _Peter Bala_, Jan 12 2016
%C A002895 This is one of the Apéry-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017
%C A002895 Named after the British-Israeli theoretical physicist Cyril Domb (1920-2012). - _Amiram Eldar_, Mar 20 2021
%D A002895 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002895 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002895 Indranil Ghosh, <a href="/A002895/b002895.txt">Table of n, a(n) for n = 0..832</a> (terms 0..100 from T. D. Noe)
%H A002895 B. Adamczewski, Jason P. Bell and E. Delaygue, <a href="http://arxiv.org/abs/1603.04187">Algebraic independence of G-functions and congruences "a la Lucas"</a>, arXiv preprint arXiv:1603.04187 [math.NT], 2016.
%H A002895 David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, <a href="https://doi.org/10.1088/1751-8113/41/20/205203">Elliptic integral evaluations of Bessel moments and applications</a>, Journal of Physics A: Mathematical and Theoretical, Vol. 41, No. 20 (2008), 205203; <a href="http://arxiv.org/abs/0801.0891">arXiv preprint</a>, arXiv:0801.0891 [hep-th], 2008.
%H A002895 Nikolai Beluhov, <a href="https://arxiv.org/abs/2506.12789">Powers of 2 in High-Dimensional Lattice Walks</a>, arXiv:2506.12789 [math.CO], 2025. See p. 20.
%H A002895 Jonathan M. Borwein, <a href="http://doi.org/10.5642/jhummath.201601.07">A short walk can be beautiful</a>, Journal of Humanistic Mathematics, Vol. 6, No. 1 (2016), pp. 86-109; <a href="https://carmamaths.org/resources/jon/beauty.pdf">preprint</a>, 2015.
%H A002895 Jonathan M. Borwein, <a href="https://carmamaths.org/resources/jon/OEIStalk.pdf">Adventures with the OEIS: Five sequences Tony may like</a>, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
%H A002895 Jonathan M. Borwein, <a href="/A060997/a060997.pdf">Adventures with the OEIS: Five sequences Tony may like</a>, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
%H A002895 Jonathan M. Borwein and Armin Straub, <a href="https://doi.org/10.1016/j.tcs.2012.10.025">Mahler measures, short walks and log-sine integrals</a>, Theoretical Computer Science, Vol. 479 (2013), pp. 4-21.
%H A002895 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 A002895 Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, <a href="https://carmamaths.org/resources/jon/Preprints/Papers/Submitted%20Papers/Walks/fpsac.pdf">Random Walk Integrals</a>, 2010.
%H A002895 Alin Bostan, Andrew Elvey Price, Anthony John Guttmann and Jean-Marie Maillard, <a href="https://arxiv.org/abs/2001.00393">Stieltjes moment sequences for pattern-avoiding permutations</a>, arXiv:2001.00393 [math.CO], 2020.
%H A002895 H. Huat Chan, Song Heng Chan and Zhiguo Liu, <a href="http://dx.doi.org/10.1016/j.aim.2003.07.012">Domb's numbers and Ramanujan-Sato type series for 1/pi</a>, Adv. Math., Vol. 186, No. 2 (2004), pp. 396-410.
%H A002895 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. See Table 2 p. 7.
%H A002895 Shaun Cooper, James G. Wan and Wadim Zudilin, <a href="https://doi.org/10.1007/978-3-319-68376-8_12">Holonomic Alchemy and Series for 1/pi</a>, in: G. Andrews and F. Garvan (eds.) Analytic Number Theory, Modular Forms and q-Hypergeometric Series, ALLADI60 2016, Springer Proceedings in Mathematics & Statistics, Vol 221. Springer, Cham, 2016; <a href="https://arxiv.org/abs/1512.04608">arXiv preprint</a>, arXiv:1512.04608 [math.NT], 2015.
%H A002895 Eric Delaygue, <a href="https://doi.org/10.1112/S0010437X17007552">Arithmetic properties of Apéry-like numbers</a>, Compositio Mathematica, Vol. 154, No. 2 (2018), pp. 249-274; <a href="http://arxiv.org/abs/1310.4131">arXiv preprint</a>, arXiv:1310.4131 [math.NT], 2013-2015.
%H A002895 Cyril Domb, <a href="http://dx.doi.org/10.1080/00018736000101199">On the theory of cooperative phenomena in crystals</a>, Advances in Phys., Vol. 9 (1960), pp. 149-361.
%H A002895 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 alpha p. 3.
%H A002895 John 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, Vol. 51 (1995), pp. 291-313.
%H A002895 Timothy Huber, Daniel Schultz, and Dongxi Ye, <a href="https://doi.org/10.4064/aa220621-19-12">Ramanujan-Sato series for 1/pi</a>, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
%H A002895 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 A002895 Ji-Cai Liu, <a href="https://arxiv.org/abs/2008.02647">Supercongruences for sums involving Domb numbers</a>, arXiv:2008.02647 [math.NT], 2020.
%H A002895 Rui-Li Liu and Feng-Zhen Zhao, <a href="https://www.emis.de/journals/JIS/VOL21/Liu/liu19.html">New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
%H A002895 Yen Lee Loh, <a href="https://doi.org/10.1088/1751-8121/aa85f6">A general method for calculating lattice green functions on the branch cut</a>, Journal of Physics A: Mathematical and Theoretical, Vol. 50, No. 40 (2017), 405203; <a href="https://arxiv.org/abs/1706.03083">arXiv preprint</a>, arXiv:1706.03083 [math-ph], 2017.
%H A002895 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 A002895 Guo-Shuai Mao and Yan Liu, <a href="https://arxiv.org/abs/2112.00511">Proof of some conjectural congruences involving Domb numbers</a>, arXiv:2112.00511 [math.NT], 2021.
%H A002895 Guo-Shuai Mao and Michael J. Schlosser, <a href="https://arxiv.org/abs/2112.12732">Supercongruences involving Domb numbers and binary quadratic forms</a>, arXiv:2112.12732 [math.NT], 2021.
%H A002895 Robert Osburn and Brundaban Sahu, <a href="https://projecteuclid.org/euclid.facm/1364222827">A supercongruence for generalized Domb numbers</a>, Functiones et Approximatio Commentarii Mathematici, Vol. 48, No. 1 (2013), pp. 29-36; <a href="http://maths.ucd.ie/~osburn/superdomb.pdf">preprint</a>.
%H A002895 L. B. Richmond and Jeffrey Shallit, <a href="http://ftp.gwdg.de/pub/EMIS/journals/EJC/Volume_16/PDF/v16i1r72.pdf">Counting Abelian Squares</a>, The Electronic Journal of Combinatorics, Vol. 16, No. 1 (2009), Article R72; <a href="http://arxiv.org/abs/0807.5028">arXiv preprint</a>, arXiv:0807.5028 [math.CO], 2008.
%H A002895 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.
%H A002895 Zhi-Hong Sun, <a href="https://arxiv.org/abs/2002.12072">Super congruences concerning binomial coefficients and Apéry-like numbers</a>, arXiv:2002.12072 [math.NT], 2020.
%H A002895 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 A002895 Zhi-Wei Sun, <a href="https://doi.org/10.1142/9789814452458_0014">Conjectures involving arithmetical sequences</a>, in: S. Kanemitsu, H. Li and J. Liu (eds.), Number Theory: Arithmetic in Shangri-La, Proc. the 6th China-Japan Sem. Number Theory (Shanghai, August 15-17, 2011), World Sci., Singapore, 2013, pp. 244-258; <a href="http://maths.nju.edu.cn/~zwsun/143p.pdf">alternative link</a>.
%H A002895 H. A. Verrill, <a href="https://arxiv.org/abs/math/0407327">Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations</a>, arXiv:math/0407327 [math.CO], 2004.
%H A002895 Chen Wang, <a href="https://arxiv.org/abs/2003.09888">Supercongruences and hypergeometric transformations</a>, arXiv:2003.09888 [math.NT], 2020.
%H A002895 Yi Wang and BaoXuan Zhu, <a href="https://doi.org/10.1007/s11425-014-4851-x">Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences</a>, Science China Mathematics, Vol. 57, No. 11 (2014), pp. 2429-2435; <a href="http://arxiv.org/abs/1303.5595">arXiv preprint</a>, arXiv:1303.5595 [math.CO], 2013.
%H A002895 Bao-Xuan Zhu, <a href="http://arxiv.org/abs/1309.6025">Higher order log-monotonicity of combinatorial sequences</a>, arXiv preprint, arXiv:1309.6025 [math.CO], 2013.
%F A002895 a(n) = Sum_{k=0..n} binomial(n, k)^2 * binomial(2n-2k, n-k) * binomial(2k, k).
%F A002895 D-finite with recurrence: n^3*a(n) = 2*(2*n-1)*(5*n^2-5*n+2)*a(n-1) - 64*(n-1)^3*a(n-2). - _Vladeta Jovovic_, Jul 16 2004
%F A002895 Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0, 2*sqrt(x))^4. - _Vladeta Jovovic_, Aug 01 2006
%F A002895 G.f.: hypergeom([1/6, 1/3],[1],108*x^2/(1-4*x)^3)^2/(1-4*x). - _Mark van Hoeij_, Oct 29 2011
%F A002895 From _Zhi-Wei Sun_, Mar 20 2013: (Start)
%F A002895 Via the Zeilberger algorithm, _Zhi-Wei Sun_ proved that:
%F A002895 (1) 4^n*a(n) = Sum_{k = 0..n} (binomial(2k,k)*binomial(2(n-k),n-k))^3/ binomial(n,k)^2,
%F A002895 (2) a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*binomial(2k,n)*binomial(2k,k)* binomial(2(n-k),n-k). (End)
%F A002895 a(n) ~ 2^(4*n+1)/((Pi*n)^(3/2)). - _Vaclav Kotesovec_, Aug 20 2013
%F A002895 G.f. y=A(x) satisfies: 0 = x^2*(4*x - 1)*(16*x - 1)*y''' + 3*x*(128*x^2 - 30*x + 1)*y'' + (448*x^2 - 68*x + 1)*y' + 4*(16*x - 1)*y. - _Gheorghe Coserea_, Jun 26 2018
%F A002895 a(n) = Sum_{p+q+r+s=n} (n!/(p!*q!*r!*s!))^2 with p,q,r,s >= 0. See Verrill, p. 5. - _Peter Bala_, Jan 06 2020
%F A002895 From _Peter Bala_, Jul 25 2024: (Start)
%F A002895 a(n) = 2*Sum_{k = 1..n} (k/n)*binomial(n, k)^2*binomial(2*n-2*k, n-k)* binomial(2*k, k) for n >= 1.
%F A002895 a(n-1) = (1/2)*Sum_{k = 1..n} (k/n)^3*binomial(n, k)^2*binomial(2*n-2*k, n-k)* binomial(2*k, k) for n >= 1. Cf. A081085. (End)
%p A002895 A002895 := n -> add(binomial(n,k)^2*binomial(2*n-2*k,n-k)*binomial(2*k,k), k=0..n): seq(A002895(n), n=0..25); # _Wesley Ivan Hurt_, Dec 20 2015
%p A002895 A002895 := n -> binomial(2*n,n)*hypergeom([1/2, -n, -n, -n],[1, 1, 1/2 - n], 1):
%p A002895 seq(simplify(A002895(n)), n=0..19); # _Peter Luschny_, May 23 2017
%t A002895 Table[Sum[Binomial[n,k]^2 Binomial[2n-2k,n-k]Binomial[2k,k],{k,0,n}], {n,0,30}] (* _Harvey P. Dale_, Aug 15 2011 *)
%t A002895 a[n_] = Binomial[2*n, n]*HypergeometricPFQ[{1/2, -n, -n, -n}, {1, 1, 1/2-n}, 1]; (* or *) a[n_] := SeriesCoefficient[BesselI[0, 2*Sqrt[x]]^4, {x, 0, n}]*n!^2; Table[a[n], {n, 0, 19}] (* _Jean-François Alcover_, Dec 30 2013, after _Vladeta Jovovic_ *)
%t A002895 max = 19; Total /@ MatrixPower[Table[Binomial[n, k]^2, {n, 0, max}, {k, 0, max}], 3] (* _Jean-François Alcover_, Mar 24 2015, after _Peter Bala_ *)
%o A002895 (PARI) C=binomial;
%o A002895 a(n) = sum(k=0,n, C(n,k)^2 * C(2*n-2*k,n-k) * C(2*k,k) );
%o A002895 /* _Joerg Arndt_, Apr 19 2013 */
%o A002895 (Python)
%o A002895 from math import comb
%o A002895 def A002895(n): return (sum(comb(n,k)**2*comb(n-k<<1,n-k)*comb(m:=k<<1,k) for k in range(n+1>>1))<<1) + (0 if n&1 else comb(n,n>>1)**4) # _Chai Wah Wu_, Jun 17 2025
%Y A002895 Cf. A002893, A008459, A169714, A169715, A228289, A267219.
%Y A002895 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 A002895 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 A002895 nonn,easy,nice,walk
%O A002895 0,2
%A A002895 _N. J. A. Sloane_
%E A002895 More terms from _Vladeta Jovovic_, Mar 11 2003