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A002895 Domb numbers: number of 2n-step polygons on diamond lattice.
(Formerly M3626 N1473)
14
1, 4, 28, 256, 2716, 31504, 387136, 4951552, 65218204, 878536624, 12046924528, 167595457792, 2359613230144, 33557651538688, 481365424895488, 6956365106016256, 101181938814289564, 1480129751586116848, 21761706991570726096, 321401321741959062016 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

Row sums of the cube of A008459. - Peter Bala, Mar 05 2013

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.

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

This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017

REFERENCES

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

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

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..832 (terms 0..100 from T. D. Noe)

B Adamczewski, JP Bell, E Delaygue, Algebraic independence of G-functions and congruences "a la Lucas", arXiv preprint arXiv:1603.04187, 2016

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008.

J. M. Borwein, A short walk can be beautiful, 2015.

Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals (2012).

Jonathan M. Borwein, Armin Straub and Christophe Vignat, Densities of short uniform random walks, Part II: Higher dimensions, Preprint, 2015.

Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Random Walk Integrals, 2010.

H. Huat Chan, Song Heng Chan, Zhiguo Liu, Domb's numbers and Ramanujan-Sato type series for 1/pi, Adv. Math. 186 (2004) 396.

Shaun Cooper, J. G. Wan and W. Zudilin, Holonomic alchemy and series for 1/pi, arXiv preprint arXiv:1512.04608 [math.NT], 2015.

E. Delaygue, Arithmetic properties of Apery-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013-2015.

C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.

J. A. Hendrickson, Jr., On the enumeration of rectangular (0,1)-matrices, Journal of Statistical Computation and Simulation, 51 (1995), 291-313.

Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5

Robert Osburn and Brundaban Sahu, A supercongruence for generalized Domb numbers.

L. B. Richmond, J. Shallit, Counting Abelian Squares, arXiv:0807.5028 [math.CO], 2008.

Armin Straub, Arithmetic aspects of random walks and methods in definite integration, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012.

Z.-W. Sun, Conjectures involving arithmetical sequences, Number Theory: Arithmetic in Shangri-La (eds., S. Kanemitsu, H.-Z. Li and J.-Y. Liu), Proc. the 6th China-Japan Sem. Number Theory (Shanghai, August 15-17, 2011), World Sci., Singapore, 2013, pp. 244-258. [broken link]

Yi Wang and Bao-Xuan Zhu, Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences, arXiv preprint arXiv:1303.5595 [math.CO], 2013.

Bao-Xuan Zhu, Higher order log-monotonicity of combinatorial sequences, arXiv preprint arXiv:1309.6025 [math.CO], 2013.

FORMULA

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

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

Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0, 2*sqrt(x))^4. - Vladeta Jovovic, Aug 01 2006

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.

From Zhi-Wei Sun, Mar 20 2013: (Start)

Via the Zeilberger algorithm, Zhi-Wei Sun proved that:

(1) 4^n*a(n) = Sum_{k = 0..n} (binomial(2k,k)*binomial(2(n-k),n-k))^3/ binomial(n,k)^2,

(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)

a(n) ~ 2^(4*n+1)/((Pi*n)^(3/2)). - Vaclav Kotesovec, Aug 20 2013

MAPLE

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

A002895 := n -> binomial(2*n, n)*hypergeom([1/2, -n, -n, -n], [1, 1, 1/2 - n], 1):

seq(simplify(A002895(n)), n=0..19); # Peter Luschny, May 23 2017

MATHEMATICA

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 *)

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 *)

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 *)

PROG

(PARI) C=binomial;

a(n) = sum(k=0, n, C(n, k)^2 * C(2*n-2*k, n-k) * C(2*k, k) );

/* Joerg Arndt, Apr 19 2013 */

CROSSREFS

Cf. A002893, A008459, A169714, A169715, A228289, A267219.

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.)

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.

Sequence in context: A230640 A191801 A064340 * A141004 A217806 A152410

Adjacent sequences:  A002892 A002893 A002894 * A002896 A002897 A002898

KEYWORD

nonn,easy,nice,walk,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Mar 11 2003

STATUS

approved

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Last modified August 22 20:47 EDT 2017. Contains 290951 sequences.