A086231 -id:A086231 - cp's OEIS frontend

A245672 Decimal expansion of k_3 = 3/(2Pim_3), a constant associated with the asymptotic expansion of the probability that a three-dimensional random walk reaches a given point for the first time, where m_3 is A086231 (Watson's integral).

3, 1, 4, 8, 7, 0, 2, 3, 1, 3, 5, 9, 6, 2, 0, 1, 7, 8, 0, 7, 5, 1, 7, 3, 9, 1, 9, 4, 1, 8, 8, 0, 6, 8, 7, 7, 0, 5, 8, 9, 6, 3, 4, 2, 4, 5, 9, 0, 1, 4, 0, 5, 5, 1, 0, 8, 4, 0, 8, 0, 3, 0, 7, 2, 7, 3, 1, 0, 8, 0, 5, 9, 4, 7, 6, 1, 4, 6, 7, 3, 1, 9, 7, 9, 7, 5, 2, 0, 2, 4, 1, 2, 0, 2, 0, 4, 9, 6, 4, 0, 4, 2, 3, 4, 4

Offset: 0

Jean-François Alcover, Jul 29 2014

			0.314870231359620178075173919418806877058963424590140551084080307273108...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's Random Walk Constants, p. 324.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's MathWorld, Polya's Random Walk Constants

Cf. A086231.

k3 = 8*Sqrt[6]*Pi^2/(Gamma[1/24]*Gamma[5/24]*Gamma[7/24]*Gamma[11/24]); RealDigits[k3, 10, 105] // First

k_3 = 8*sqrt(6)*Pi^2/(Gamma(5/24)*Gamma(7/24)*Gamma(11/24)), where 'Gamma' is the Euler gamma function.

Asymptotic probability ~ k_3 / ||l||, where the norm ||l|| of the position of the lattice point l tends to infinity.

A002893 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(2*k,k).

Original entry on oeis.org

1, 3, 15, 93, 639, 4653, 35169, 272835, 2157759, 17319837, 140668065, 1153462995, 9533639025, 79326566595, 663835030335, 5582724468093, 47152425626559, 399769750195965, 3400775573443089, 29016970072920387, 248256043372999089

Offset: 0

N. J. A. Sloane

This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004
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
a(n) is the number of Abelian squares of length 2n over a 3-letter alphabet. - Jeffrey Shallit, Aug 17 2010
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
Row sums of A318397 the square of A008459. - Peter Bala, Mar 05 2013
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
This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
a(n) is the sum of the squares of the coefficients of (x + y + z)^n. - Michael Somos, Aug 25 2018
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

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

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

Seiichi Manyama, Table of n, a(n) for n = 0..1051 (terms 0..100 from T. D. Noe)
B. Adamczewski, J. P. Bell, and E. Delaygue, Algebraic independence of G-functions and congruences à la Lucas", arXiv preprint arXiv:1603.04187 [math.NT], 2016.
Rostislav Akhmechet, Mikhail Khovanov, and Melissa Zhang, Annular SL(2) and SL(3) web algebras, arXiv:2508.05605 [math.GT], 2025. See p. 48.
Gert Almkvist, The art of finding Calabi-Yau differential equations, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008.
P. Barrucand, A combinatorial identity, Problem 75-4, SIAM Rev., 17 (1975), 168. Solution by D. R. Breach, D. McCarthy, D. Monk and P. E. O'Neil, SIAM Rev. 18 (1976), 303.
P. Barrucand, Problem 75-4, A Combinatorial Identity, SIAM Rev., 17 (1975), 168. [Annotated scanned copy of statement of problem]
Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982.
Frits Beukers and Jan Stienstra, On the Picard-Fuchs Equation and the Formal Brauer Group of Certain Elliptic K3-Surfaces, Mathematische Annalen (1985), Vol. 271, pp. 269-304 (see Part III).
Artur Bille, Victor Buchstaber, Simon Coste, Satoshi Kuriki, and Evgeny Spodarev, Random eigenvalues of graphenes and the triangulation of plane, arXiv:2306.01462 [math.SP], 2023.
Jonathan M. Borwein, A short walk can be beautiful, 2015.
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Random Walk Integrals, 2010.
Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals.
Jonathan M. Borwein, Armin Straub and Christophe Vignat, Densities of short uniform random walks, Part II: Higher dimensions, Preprint, 2015
Jonathan M. Borwein, Armin Straub and James Wan, Three-Step and Four-Step Random Walk Integrals, Exper. Math., 22 (2013), 1-14.
Charles Burnette and Chung Wong, Abelian Squares and Their Progenies, arXiv:1609.05580 [math.CO], 2016.
David Callan, A combinatorial interpretation for an identity of Barrucand, JIS 11 (2008) 08.3.4.
Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023.
M. Coster, Email, Nov 1990
Eric Delaygue, Arithmetic properties of Apery-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013.
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
Jeffrey S. Geronimo, Hugo J. Woerdeman, and Chung Y. Wong, The autoregressive filter problem for multivariable degree one symmetric polynomials, arXiv:2101.00525 [math.CA], 2021.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See C p. 2.
Victor J. W. Guo, Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers, arXiv preprint arXiv:1201.0617 [math.NT], 2012.
Victor J. W. Guo, Guo-Shuai Mao and Hao Pan, Proof of a conjecture involving Sun polynomials, arXiv preprint arXiv:1511.04005 [math.NT], 2015.
E. Hallouin and M. Perret, A Graph Aided Strategy to Produce Good Recursive Towers over Finite Fields, arXiv preprint arXiv:1503.06591 [math.NT], 2015.
J. A. Hendrickson, Jr., On the enumeration of rectangular (0,1)-matrices, Journal of Statistical Computation and Simulation, 51 (1995), 291-313.
S. Herfurtner, Elliptic surfaces with four singular fibres, Mathematische Annalen, 1991. Preprint.
Pakawut Jiradilok and Elchanan Mossel, Gaussian Broadcast on Grids, arXiv:2402.11990 [cs.IT], 2024. See p. 27.
Davidson Noby Joseph and Igor Boettcher, Walking on Archimedean Lattices: Insights from Bloch Band Theory, arXiv:2507.12662 [cond-mat.stat-mech], 2025. See p. 18.
Tanya Khovanova and Konstantin Knop, Coins of three different weights, arXiv:1409.0250 [math.HO], 2014.
Murray S. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 148-149.
Bradley Klee, Checking Weierstrass data, 2023.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5
Mathematics Stack Exchange, sum involving the product of binomial coefficients, Nov 10 2016.
L. B. Richmond and Jeffrey Shallit, Counting abelian squares, Electronic J. Combinatorics 16 (1), #R72, June 2009. [From _Jeffrey Shallit_, Aug 17 2010]
Armin Straub, Arithmetic aspects of random walks and methods in definite integration, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012. - From _N. J. A. Sloane_, Dec 16 2012
Zhi-Hong Sun, Congruences for Apéry-like numbers, arXiv:1803.10051 [math.NT], 2018.
Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020.
Zhi-Wei Sun, Connections between p = x^2+3y^2 and Franel numbers, J. Number Theory 133(2013), 2919-2928.
Zhi-Wei Sun, Congruences involving g_n(x) = Sum_{k=0..n} binomial(n,k)^2*binomial(2k,k)*x^k, Ramanujan J., in press. Doi: 10.1007/s11139-015-9727-3.
Brani Vidakovic, All roads lead to Rome--even in the honeycomb world, Amer. Statist., 48 (1994) no. 3, 234-236.
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.
D. Zagier, Integral solutions of Apery-like recurrence equations. See line C in sporadic solutions table of page 5.

Cf. A000172, A002895, A000984, A006480, A087457, A274600, A318397, A244973.
Cf. A169714, A169715.
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.

[&+[Binomial(n, k)^2 * Binomial(2*k, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 26 2018

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
A002893 := n -> hypergeom([1/2, -n, -n], [1, 1], 4):
seq(simplify(A002893(n)), n=0..20); # Peter Luschny, May 23 2017

Table[Sum[Binomial[n,k]^2 Binomial[2k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Aug 19 2011 *)
a[ n_] := If[ n < 0, 0, HypergeometricPFQ[ {1/2, -n, -n}, {1, 1}, 4]]; (* Michael Somos, Oct 16 2013 *)
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 *)
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 *)

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

{a(n) = sum(k=0, n, binomial(n, k)^2 * binomial(2*k, k))}; /* Michael Somos, Jul 25 2007 */

{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

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

def A002893(n): return simplify(hypergeometric([1/2,-n,-n], [1,1], 4))
[A002893(n) for n in range(31)] # G. C. Greubel, Jan 21 2023

a(n) = Sum_{m=0..n} binomial(n, m) * A000172(m). [Barrucand]
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
Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0, 2*sqrt(x))^3. - Vladeta Jovovic, Mar 11 2003
a(n) = Sum_{p+q+r=n} (n!/(p!*q!*r!))^2 with p, q, r >= 0. - Michael Somos, Jul 25 2007
a(n) = 3*A087457(n) for n>0. - Philippe Deléham, Sep 14 2008
a(n) = hypergeom([1/2, -n, -n], [1, 1], 4). - Mark van Hoeij, Jun 02 2010
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
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
Asymptotic: a(n) ~ 3^(2*n+3/2)/(4*Pi*n). - Vaclav Kotesovec, Sep 11 2012
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
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
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
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
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
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
From Bradley Klee, Jun 05 2023: (Start)
The g.f. T(x) obeys a period-annihilating ODE:
0=3*(-1 + 3*x)*T(x) + (1 - 20*x + 27*x^2)*T'(x) + x*(-1 + x)*(-1 + 9*x)*T''(x).
The periods ODE can be derived from the following Weierstrass data:
g2 = (3/64)*(1 + 3*x)*(1 - 15*x + 75*x^2 + 3*x^3);
g3 = -(1/512)*(-1 + 6*x + 3*x^2)*(1 - 12*x + 30*x^2 - 540*x^3 + 9*x^4);
which determine an elliptic surface with four singular fibers. (End)
a(n) = Sum_{k = 0..n} binomial(n, k)^2 * binomial(3*k, 2*n) (Almkvist, p. 16). - Peter Bala, May 22 2025

A002896 Number of 2n-step polygons on cubic lattice.

Original entry on oeis.org

1, 6, 90, 1860, 44730, 1172556, 32496156, 936369720, 27770358330, 842090474940, 25989269017140, 813689707488840, 25780447171287900, 825043888527957000, 26630804377937061000, 865978374333905289360, 28342398385058078078010, 932905175625150142902300

Offset: 0

N. J. A. Sloane

Number of walks with 2n steps on the cubic lattice Z^3 beginning and ending at (0,0,0).
If A is a random matrix in USp(6) (6 X 6 complex matrices that are unitary and symplectic) then a(n) is the 2n-th moment of tr(A^k) for all k >= 7. - Andrew V. Sutherland, Mar 24 2008
Diagonal of the rational function R(x,y,z,w) = 1/(1 - (w*x*y + w*x*z + w*y + x*z + y + z)). - Gheorghe Coserea, Jul 14 2016
Constant term in the expansion of (x + 1/x + y + 1/y + z + 1/z)^(2n). - Harry Richman, Apr 29 2020

			1 + 6*x + 90*x^2 + 1860*x^3 + 44730*x^4 + 1172556*x^5 + 32496156*x^6 + ...

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).

Alois P. Heinz, Table of n, a(n) for n = 0..645
David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008.
Nikolai Beluhov, Powers of 2 in Balanced Grid Colourings, arXiv:2504.21451 [math.CO], 2025.
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024. See p. 5.
Li Gan, On the algebraic area of cubic lattice walks, arXiv:2307.08732 [math-ph], 2023.
S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 23.
W. K. Hayman, A Generalisation of Stirling's Formula, Journal für die reine und angewandte Mathematik, (1956), vol. 196, pp. 67-95.
John A. Hendrickson, Jr., On the enumeration of rectangular (0,1)-matrices, Journal of Statistical Computation and Simulation, 51 (1995), 291-313.
Heng Huat Chan, Yoshio Tanigawa, Yifan Yang, and Wadim Zudilin, New analogues of Clausen's identities arising from the theory of modular forms, Advances in Mathematics, 228:2 (2011), pp. 1294-1314; doi:10.1016/j.aim.2011.06.011.
G. S. Joyce, The simple cubic lattice Green function, Phil. Trans. Roy. Soc., 273 (1972), 583-610.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017.
Nobu C. Shirai and Naoyuki Sakumichi, Universal Negative Energetic Elasticity in Polymer Chains: Crossovers among Random, Self-Avoiding, and Neighbor-Avoiding Walks, arXiv:2408.14992 [cond-mat.soft], 2024. See p. 5.
Chunwei Song and Bowen Yao, On Combinatorial Rectangles with Minimum oo-Discrepancy, arXiv:1909.05648 [math.CO], 2019. See p. 7 for another interpretation.
Ralph A. Raimi and Jet Wimp, Review of book "A=B" by M. Petkovsek et al., Mathematical Intelligencer, 23 (No. 4, 2001), pp. 72-77.

C(2n, n) times A002893.
Cf. A049020, A049037, A084261, A138540, A174516.
Related to diagonal of rational functions: A268545-A268555.
Row k=3 of A287318.

a := proc(n) local k; binomial(2*n,n)*add(binomial(n,k)^2 *binomial(2*k,k), k=0..n); end;
# second Maple program
a:= proc(n) option remember; `if`(n<2, 5*n+1,
      (2*(2*n-1)*(10*n^2-10*n+3) *a(n-1)
       -36*(n-1)*(2*n-1)*(2*n-3) *a(n-2)) /n^3)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 02 2012
A002896 := n -> binomial(2*n,n)*hypergeom([1/2, -n, -n], [1, 1], 4):
seq(simplify(A002896(n)), n=0..16); # Peter Luschny, May 23 2017

Table[Binomial[2n,n] Sum[Binomial[n,k]^2 Binomial[2k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jan 24 2012 *)
a[ n_] := If[ n < 0, 0, HypergeometricPFQ[ {-n, -n, 1/2}, {1, 1}, 4] Binomial[ 2 n, n]] (* Michael Somos, May 21 2013 *)

a(n)=binomial(2*n,n)*sum(k=0,n,binomial(n, k)^2*binomial(2*k, k)) \\ Charles R Greathouse IV, Oct 31 2011

def A002896():
    x, y, n = 1, 6, 1
    while True:
        yield x
        n += 1
        x, y = y, ((4*n-2)*((10*(n-1)*n+3)*y-18*(n-1)*(2*n-3)*x))//n^3
a = A002896()
[next(a) for i in range(17)]  # Peter Luschny, Oct 09 2013

a(n) = C(2*n, n)*Sum_{k=0..n} C(n, k)^2*C(2*k, k).
a(n) = (4^n*p(1/2, n)/n!)*hypergeom([-n, -n, 1/2], [1, 1], 4), where p(a, k) = Product_{i=0..k-1} (a+i).
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2*x)^3. - Corrected by Christopher J. Smyth, Oct 29 2012
D-finite with recurrence: n^3*a(n) = 2*(2*n-1)*(10*n^2-10*n+3)*a(n-1) - 36*(n-1)*(2*n-1)*(2*n-3)*a(n-2). - Vladeta Jovovic, Jul 16 2004
An asymptotic formula follows immediately from an observation of Bruce Richmond and me in SIAM Review - 31 (1989, 122-125). We use Hayman's method to find the asymptotic behavior of the sum of squares of the multinomial coefficients multi(n, k_1, k_2, ..., k_m) with m fixed. From this one gets a_n ~ (3/4)*sqrt(3)*6^(2*n)/(Pi*n)^(3/2). - Cecil C Rousseau (ccrousse(AT)memphis.edu), Mar 14 2006
G.f.: (1/sqrt(1+12*z)) * hypergeom([1/8,3/8],[1],64/81*z*(1+sqrt(1-36*z))^2*(2+sqrt(1-36*z))^4/(1+12*z)^4) * hypergeom([1/8, 3/8],[1],64/81*z*(1-sqrt(1-36*z))^2*(2-sqrt(1-36*z))^4/(1+12*z)^4). - Sergey Perepechko, Jan 26 2011
a(n) = binomial(2*n,n)*A002893(n). - Mark van Hoeij, Oct 29 2011
G.f.: (1/2)*(10-72*x-6*(144*x^2-40*x+1)^(1/2))^(1/2)*hypergeom([1/6, 1/3],[1],54*x*(108*x^2-27*x+1+(9*x-1)*(144*x^2-40*x+1)^(1/2)))^2. - Mark van Hoeij, Nov 12 2011
PSUM transform is A174516. - Michael Somos, May 21 2013
0 = (-x^2+40*x^3-144*x^4)*y''' + (-3*x+180*x^2-864*x^3)*y'' + (-1+132*x-972*x^2)*y' + (6-108*x)*y, where y is the g.f. - Gheorghe Coserea, Jul 14 2016
a(n) = [(x y z)^0] (x + 1/x + y + 1/y + z + 1/z)^(2*n). - Christopher J. Smyth, Sep 25 2018
a(n) = (1/Pi)^3*Integral_{0 <= x, y, z <= Pi} (2*cos(x) + 2*cos(y) + 2*cos(z))^(2*n) dx dy dz. - Peter Bala, Feb 10 2022
a(n) = Sum_{i+j+k=n, 0<=i,j,k<=n} multinomial(2n [i,i,j,j,k,k]). - Shel Kaphan, Jan 16 2023
Sum_{k>=0} a(k)/36^k = A086231 = (sqrt(3)-1) * (Gamma(1/24) * Gamma(11/24))^2 / (32*Pi^3). - Vaclav Kotesovec, Apr 23 2023
G.f.: HeunG(1/9,1/12,1/4,3/4,1,1/2,4*x)^2 (see Hassani et al.). - Stefano Spezia, Feb 16 2025

A086230 Decimal expansion of probability that a random walk on a 3-D lattice returns to the origin.

Original entry on oeis.org

3, 4, 0, 5, 3, 7, 3, 2, 9, 5, 5, 0, 9, 9, 9, 1, 4, 2, 8, 2, 6, 2, 7, 3, 1, 8, 4, 4, 3, 2, 9, 0, 2, 8, 9, 6, 7, 1, 0, 6, 0, 8, 2, 1, 7, 1, 2, 4, 3, 0, 2, 0, 9, 7, 7, 6, 3, 2, 3, 6, 1, 0, 5, 3, 7, 7, 7, 9, 1, 9, 6, 9, 4, 5, 8, 9, 6, 2, 3, 8, 4, 6, 4, 2, 5, 2, 8, 0, 8, 1, 8, 8, 9, 0, 5, 7, 1, 3, 0, 9, 9, 4

Offset: 0

Eric W. Weisstein, Jul 12 2003

Pólya (1921) proved that this constant is < 1. McCrea and Whipple (1940) evaluated it by 0.34. - Amiram Eldar, Aug 28 2020

			0.340537329550999142826273184432902896710608217124302097763236105377791969...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 322-331.

G. C. Greubel, Table of n, a(n) for n = 0..10000
W. H. McCrea and F. J. W. Whipple, Random Paths in Two and Three Dimensions, Proceedings of the Royal Society of Edinburgh, Vol. 60, No. 3 (1940), pp. 281-298. See p. 297.
Georg Pólya, Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz, Mathematische Annalen, Vol. 84, No. 1-2 (1921), pp. 149-160.
Eric Weisstein's World of Mathematics, Pólya's Random Walk Constants.

Cf. A086231, A086232, A086233, A086234, A086235, A086236.

C := ComplexField(); 1 - (16*Sqrt(2/3)*Pi(C)^3)/(Gamma(1/24)* Gamma(5/24)*Gamma(7/24)*Gamma(11/24)); // G. C. Greubel, Jan 25 2018

Mathematica

RealDigits[1 - (16*Sqrt[2/3]*Pi^3) / (Gamma[1/24]*Gamma[5/24]*Gamma[7/24]*Gamma[11/24]), 10, 102] // First (* Jean-François Alcover, Feb 08 2013, after Eric W. Weisstein *)

PARI

1-32*Pi^3/sqrt(6)/gamma(1/24)/gamma(5/24)/gamma(7/24)/gamma(11/24) \\ Charles R Greathouse IV, Jul 22 2013

Formula

Equals 1 - (16*Sqrt(2/3)*Pi^3)/(Gamma(1/24)* Gamma(5/24)*Gamma(7/24)* Gamma(11/24)). - G. C. Greubel, Jan 25 2018

Equals 1 - 1/A086231. - Amiram Eldar, Aug 28 2020

A242812 Decimal expansion of the expected number of returns to the origin of a random walk on a 4-d lattice.

Original entry on oeis.org

1, 2, 3, 9, 4, 6, 7, 1, 2, 1, 8, 4, 8, 4, 8, 1, 7, 1, 2, 6, 7, 8, 6, 9, 7, 6, 6, 4, 8, 5, 9, 0, 0, 0, 7, 1, 0, 1, 5, 3, 2, 8, 9, 0, 6, 9, 1, 6, 1, 7, 5, 8, 6, 5, 6, 9, 5, 3, 4, 0, 1, 8, 5, 0, 7, 1, 6, 2, 8, 1, 3, 3, 8, 6, 5, 5, 5, 6, 3, 3, 3, 1, 0, 3, 2, 3, 9, 3, 3, 0, 4, 7, 3, 5, 3, 8, 9, 3, 9, 2, 8, 5, 9, 9, 1, 8

Offset: 1

Jean-François Alcover, May 23 2014

			1.239467121848481712678697664859...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.

Marc Mezzarobba, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pólya's Random Walk Constants.

Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242813.

m4:= int(exp(-t)*BesselI(0, t/4)^4, t=0..infinity):
s:= convert(evalf(m4, 120), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014

digits = 50; NIntegrate[BesselI[0, t/4]^4*Exp[-t], {t, 0, Infinity}, PrecisionGoal -> digits, WorkingPrecision -> 350] // RealDigits [#, 10, digits]& // First (* after Ryan Propper *)

m(d) = d/(2*Pi)^d*multipleIntegral(-Pi..Pi) (d-sum_(k=1..d) cos(t_k))^(-1) dt_1 dt_2 ... dt_d, where d is the lattice dimension.
m(d) = Integral_(t>0) exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
Equals 1/(1 - A086232). - Amiram Eldar, Aug 28 2020

More terms from Alois P. Heinz, May 23 2014

A242813 Decimal expansion of the expected number of returns to the origin of a random walk on a 5-d lattice.

Original entry on oeis.org

1, 1, 5, 6, 3, 0, 8, 1, 2, 4, 8, 4, 0, 2, 3, 1, 1, 7, 8, 7, 0, 7, 1, 3, 5, 1, 2, 1, 9, 3, 8, 5, 6, 6, 9, 8, 5, 5, 4, 5, 4, 2, 7, 3, 4, 8, 5, 0, 5, 1, 4, 2, 3, 8, 8, 2, 6, 9, 5, 6, 6, 0, 1, 1, 2, 1, 0, 0, 8, 7, 7, 0, 3, 4, 7, 0, 6, 8, 7, 3, 1, 1, 7, 2, 3, 6, 6, 5, 4, 3, 0, 4, 9, 5, 0, 9, 1, 7, 1, 6, 5, 2, 6, 7, 4, 3

Offset: 1

Jean-François Alcover, May 23 2014

			1.1563081248...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.

Marc Mezzarobba, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pólya's Random Walk Constants.

Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242812, A242814, A242815, A242816.

m5:= int(exp(-t)*BesselI(0, t/5)^5, t=0..infinity):
s:= convert(evalf(m5, 120), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014

d = 5; d/Pi^d*NIntegrate[(d - Sum[Cos[t[k]], {k, 1, d}])^-1, Sequence @@ Table[{t[k], 0, Pi}, {k, 1, d}] // Evaluate] // RealDigits[#, 10, 10]& // First

intnumosc(t=0,exp(-t)*besseli(0,t/5)^5,Pi*5) \\ Charles R Greathouse IV, Oct 23 2023

m(d) = d/(2*Pi)^d*multipleIntegral(-Pi..Pi) (d-sum_(k=1..d) cos(t_k))^(-1) dt_1 dt_2 ... dt_d, where d is the lattice dimension.
m(d) = integral_(t>0) exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
Equals 1/(1 - A086233). - Amiram Eldar, Aug 28 2020

More terms from Alois P. Heinz, May 23 2014

A242814 Decimal expansion of the expected number of returns to the origin of a random walk on a 6-d lattice.

Original entry on oeis.org

1, 1, 1, 6, 9, 6, 3, 3, 7, 3, 2, 2, 6, 6, 7, 1, 8, 4, 3, 6, 8, 5, 6, 4, 4, 3, 3, 1, 9, 6, 8, 6, 1, 3, 2, 5, 2, 6, 5, 6, 1, 9, 2, 6, 2, 2, 3, 9, 3, 0, 3, 2, 5, 2, 4, 6, 8, 3, 9, 9, 9, 5, 2, 9, 4, 0, 0, 4, 5, 6, 0, 7, 6, 4, 5, 4, 7, 0, 0, 8, 7, 9, 5, 2, 3, 2, 5, 0, 5, 4, 2, 8, 5, 1, 8, 3, 5, 4, 7, 7, 7, 2, 7, 5, 7, 8

Offset: 1

Jean-François Alcover, May 23 2014

			1.1169633732...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.

Marc Mezzarobba, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pólya's Random Walk Constants.

Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242812, A242813.

m6:= int(exp(-t)*BesselI(0, t/6)^6, t=0..infinity):
s:= convert(evalf(m6, 120), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014

d = 6; d/Pi^d*NIntegrate[(d - Sum[Cos[t[k]], {k, 1, d}])^-1, Sequence @@ Table[{t[k], 0, Pi}, {k, 1, d}] // Evaluate] // RealDigits[#, 10, 8]& // First

intnumosc(t=0,exp(-t)*besseli(0,t/6)^6,12*Pi) \\ Charles R Greathouse IV, Oct 23 2023

m(d) = d/(2*Pi)^d*multipleIntegral(-Pi..Pi) (d-sum_(k=1..d) cos(t_k))^(-1) dt_1 dt_2 ... dt_d, where d is the lattice dimension.
m(d) = integral_(t>0) exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
Equals 1/(1 - A086234). - Amiram Eldar, Aug 28 2020

More terms from Alois P. Heinz, May 23 2014

A242816 Decimal expansion of the expected number of returns to the origin of a random walk on an 8-d lattice.

Original entry on oeis.org

1, 0, 7, 8, 6, 4, 7, 0, 1, 2, 0, 1, 6, 9, 2, 5, 5, 5, 8, 6, 4, 2, 6, 8, 4, 4, 8, 0, 0, 2, 7, 4, 1, 5, 0, 6, 1, 1, 5, 0, 3, 3, 1, 9, 9, 8, 7, 2, 3, 5, 3, 8, 3, 1, 1, 3, 2, 8, 1, 7, 8, 6, 8, 1, 8, 2, 4, 4, 0, 9, 1, 2, 7, 8, 9, 4, 4, 4, 5, 5, 9, 0, 8, 7, 4, 8, 0, 4, 8, 0, 7, 1, 6, 3, 2, 3, 1, 9, 0, 0, 7, 1, 0, 1, 9

Offset: 1

Jean-François Alcover, May 23 2014

			1.0786470120...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.

Marc Mezzarobba, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pólya's Random Walk Constants.

Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242812, A242813, A242814, A242815.

m8:= int(exp(-t)*BesselI(0, t/8)^8, t=0..infinity):
s:= convert(evalf(m8, 120), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014

d = 8; d/Pi^d*NIntegrate[(d - Sum[Cos[t[k]], {k, 1, d}])^-1, Sequence @@ Table[{t[k], 0, Pi}, {k, 1, d}] // Evaluate] // RealDigits[#, 10, 7]& // First

m(d) = d/(2*Pi)^d*multipleIntegral(-Pi..Pi) (d-sum_(k=1..d) cos(t_k))^(-1) dt_1 dt_2 ... dt_d, where d is the lattice dimension.
m(d) = Integral_{t>0} exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
Equals 1/(1 - A086236). - Amiram Eldar, Aug 28 2020

More terms from Alois P. Heinz, May 23 2014

A242815 Decimal expansion of the expected number of returns to the origin of a random walk on a 7-d lattice.

Original entry on oeis.org

1, 0, 9, 3, 9, 0, 6, 3, 1, 5, 5, 8, 7, 8, 4, 7, 9, 9, 6, 6, 8, 3, 2, 7, 1, 8, 2, 3, 5, 5, 9, 0, 1, 9, 8, 6, 3, 7, 1, 1, 2, 8, 9, 9, 7, 7, 1, 6, 4, 9, 6, 1, 1, 5, 4, 4, 9, 1, 6, 8, 9, 0, 7, 3, 8, 8, 6, 1, 2, 6, 5, 4, 5, 7, 0, 5, 0, 8, 0, 5, 2, 2, 8, 4, 4, 8, 9, 5, 1, 9, 1, 9, 7, 2, 9, 8, 5, 5, 9, 8, 7, 5, 7, 2, 9, 9

Offset: 1

Jean-François Alcover, May 23 2014

			1.09390631558784799668327...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.

Marc Mezzarobba, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pólya's Random Walk Constants.

Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242812, A242813, A242814, A242816.

m7:= int(exp(-t)*BesselI(0, t/7)^7, t=0..infinity):
s:= convert(evalf(m7, 120), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014

d = 7; d/Pi^d*NIntegrate[(d - Sum[Cos[t[k]], {k, 1, d}])^-1, Sequence @@ Table[{t[k], 0, Pi}, {k, 1, d}] // Evaluate] // RealDigits[#, 10, 7]& // First

m(d) = d/(2*Pi)^d*multipleIntegral(-Pi..Pi) (d-sum_(k=1..d) cos(t_k))^(-1) dt_1 dt_2 ... dt_d, where d is the lattice dimension.
m(d) = Integral_{t>0} exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
Equals 1/(1 - A086235). - Amiram Eldar, Aug 28 2020

More terms from Alois P. Heinz, May 23 2014

A273086 Decimal expansion of theta_3(0, exp(-sqrt(6)*Pi)), where theta_3 is the 3rd Jacobi theta function.

Original entry on oeis.org

1, 0, 0, 0, 9, 0, 9, 9, 2, 1, 8, 8, 7, 2, 5, 6, 7, 6, 2, 9, 1, 9, 2, 8, 6, 0, 0, 4, 1, 2, 1, 5, 6, 6, 6, 7, 1, 8, 0, 4, 5, 8, 8, 1, 4, 6, 7, 3, 0, 3, 0, 1, 3, 3, 0, 8, 5, 9, 2, 4, 1, 7, 9, 6, 8, 1, 3, 9, 5, 8, 5, 4, 2, 0, 8, 7, 9, 5, 0, 0, 5, 6, 3, 3, 2, 7, 5, 4, 2, 2, 0, 2, 2, 1, 8, 2, 9, 1, 1, 4, 7, 4, 2, 1, 8

Offset: 1

Vaclav Kotesovec, May 14 2016

			1.0009099218872567629192860041215666718045881467303013308592...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's MathWorld, Jacobi Theta Functions
Wikipedia, Theta function

Cf. A175573, A247217, A273081, A273082, A273083, A273084, A086231, A273087.

evalf(((6 + sqrt(6*(3 + 2*sqrt(2)))) * GAMMA(1/24) * GAMMA(5/24) * GAMMA(7/24) * GAMMA(11/24))^(1/4) / (2*6^(3/8)*Pi^(3/4)), 120);
evalf((4 - sqrt(2) + sqrt(6))^(1/4) * sqrt(GAMMA(1/24)*GAMMA(11/24)) / (2^(3/2)*3^(3/8)*Pi^(3/4)), 120);

RealDigits[EllipticTheta[3, 0, Exp[-Sqrt[6]*Pi]], 10, 105][[1]]
RealDigits[((6 + Sqrt[6*(3 + 2*Sqrt[2])]) * Gamma[1/24] * Gamma[5/24] * Gamma[7/24] * Gamma[11/24])^(1/4) / (2*6^(3/8)*Pi^(3/4)), 10, 105][[1]]
RealDigits[(4 - Sqrt[2] + Sqrt[6])^(1/4) * Sqrt[Gamma[1/24]*Gamma[11/24]] / (2^(3/2)*3^(3/8)*Pi^(3/4)), 10, 105][[1]]

th3(x)=1 + 2*suminf(n=1,x^n^2)
th3(exp(-sqrt(6)*Pi)) \\ Charles R Greathouse IV, Jun 06 2016

Equals ((6 + sqrt(6*(3 + 2*sqrt(2)))) * Gamma(1/24) * Gamma(5/24) * Gamma(7/24) * Gamma(11/24))^(1/4) / (2*6^(3/8)*Pi^(3/4)).

Equals (4 - sqrt(2) + sqrt(6))^(1/4) * sqrt(Gamma(1/24)*Gamma(11/24)) / (2^(3/2)*3^(3/8)*Pi^(3/4)).

cp's OEIS Frontend

A245672 Decimal expansion of k_3 = 3/(2*Pi*m_3), a constant associated with the asymptotic expansion of the probability that a three-dimensional random walk reaches a given point for the first time, where m_3 is A086231 (Watson's integral).

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A002893 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(2*k,k).

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A002896 Number of 2n-step polygons on cubic lattice.

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A086230 Decimal expansion of probability that a random walk on a 3-D lattice returns to the origin.

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A242812 Decimal expansion of the expected number of returns to the origin of a random walk on a 4-d lattice.

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A242813 Decimal expansion of the expected number of returns to the origin of a random walk on a 5-d lattice.

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A245672 Decimal expansion of k_3 = 3/(2Pim_3), a constant associated with the asymptotic expansion of the probability that a three-dimensional random walk reaches a given point for the first time, where m_3 is A086231 (Watson's integral).