A002893 -id:A002893 - cp's OEIS frontend

A262177 Decimal expansion of Q_5 = zeta(5) / (Sum_{k>=1} (-1)^(k+1) / (k^5 * binomial(2k, k))), a conjecturally irrational constant defined by an Apéry-like formula.

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

Offset: 1

Jean-François Alcover, Sep 14 2015

The similar constant Q_3 = zeta(3) / (Sum_{k>=1} (-1)^(k+1) / (k^3 * binomial(2k, k))) evaluates to 5/2.

			2.09486862201003699385024929373294163029675874856778182740127587837438...

G. C. Greubel, Table of n, a(n) for n = 1..5000
David Bailey, Jonathan Borwein, David Bradley, Experimental determination of Apéry-like identities for zeta(2n+2), arXiv:math/0505270 [math.NT], 2005.

Cf. A013663.

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.)

Q5 = Zeta[5]/Sum[(-1)^(k+1)/(k^5*Binomial[2k, k]), {k, 1, Infinity}]; RealDigits[Q5, 10, 103] // First

zeta(5)/suminf(k=1, (-1)^(k+1)/(k^5*binomial(2*k,k))) \\ Michel Marcus, Sep 14 2015

Equals 2*zeta(5)/6F5(1,1,1,1,1,1; 3/2,2,2,2,2; -1/4).

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

A002898 Number of n-step closed paths on hexagonal lattice.

Original entry on oeis.org

1, 0, 6, 12, 90, 360, 2040, 10080, 54810, 290640, 1588356, 8676360, 47977776, 266378112, 1488801600, 8355739392, 47104393050, 266482019232, 1512589408044, 8610448069080, 49144928795820, 281164160225520, 1612061452900080, 9261029179733760, 53299490722049520

Offset: 0

N. J. A. Sloane

Also, number of closed paths of length n on the honeycomb tiling.
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
From David Callan, Aug 25 2009: (Start)
a(n) = number of 2 X n matrices, entries from {1,2,3}, second row a (multiset) permutation of the first, with no constant columns. For example, a(2)=6 counts the matrices
   1 2   1 3   2 1   2 3   3 1   3 2
   2 1   3 1   1 2   3 2   1 3   2 3. (End)
Also, a(n) is the constant coefficient in the expansion of (x + 1/x + y + 1/y + x/y + y/x)^n. - Christopher J. Smyth, Sep 25 2018
a(n) is the constant term in the expansion of (-2 + (1 + x) * (1 + y) + (1 + 1/x) * (1 + 1/y))^n. - Seiichi Manyama, Oct 27 2019
a(n) is the number of paths from (0,0,0) to (n,n,n) using the six permutations of (0,1,2) as steps, i.e., the steps (0,1,2), (0,2,1), (1,0,2), (1,2,0), (2,0,1), and (2,1,0). - William J. Wang, Dec 07 2020

			O.g.f.: 1 + 6*x^2 + 12*x^3 + 90*x^4 + 360*x^5 + 2040*x^6 + ...
O.g.f.: 1 + 6*x^2*(1+2*x) + 90*x^4*(1+2*x)^2 + 1680*x^6*(1+2*x)^3 + 34650*x^8*(1+2*x)^4 + ... + A006480(n)*x^(2*n)*(1+2*x)^n + .... - _Paul D. Hanna_, Feb 26 2012

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).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Cyril Banderier, Analytic combinatorics of random walks and planar maps, PhD Thesis, 2001 (>6 Mb).
Volker Braun, Philip Candelas, and Xenia de la Ossa, Two One-Parameter Special Geometries, arXiv preprint arXiv:1512.08367 [hep-th], 2015.
Gunther Cornelissen, David Hokken, and Berend Ringeling, The asymptotic Mahler measure of Gaussian periods, arXiv:2507.09303 [math.NT], 2025. See p. 40.
Cyril Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
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.
Leonard F. Klosinski, Gerald L. Alexanderson and Loren C. Larson, Solution to 1995 Putnam problem A-6, Am. Math. Monthly, 1996, p. 674.
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavík, Iceland DMTCS proc. AO, 2011, 599-610.
Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017.
Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Grzegorz Siudem and Agata Fronczak, Bell polynomials in the series expansions of the Ising model, arXiv:2007.16132 [math-ph], 2020.

Cf. A000172, A006480, A337905-A337907, A094060, A002894 (returns square lattice), A002893 (honeycomb net).

a:= proc(n) option remember; `if`(n<3, [1, 0, 6][n+1], ((n-1)*
      n*a(n-1) +24*(n-1)^2*a(n-2) +36*(n-1)*(n-2)*a(n-3))/n^2)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 08 2020

a[n_] := Sum[(-2)^(n-i)*Binomial[i, j]^3*Binomial[n, i], {i, 0, n}, {j, 0, i}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Dec 21 2011, after Vasu Tewari *)

{a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3*x^(2*m)*(1+2*x+x*O(x^n))^m),n)} /* Paul D. Hanna, Feb 26 2012 */

D-finite with recurrence a(0) = 1, a(1) = 0, a(2) = 6, 36*(n+2)*(n+1)*a(n) +24*(n+2)^2*a(n+1) +(n+3)*(n+2)*a(n+2) -(n+3)^2*a(n+3) = 0.
E.g.f.: (BesselI(0,2*x))^3 + 2*Sum_{k>=1} (BesselI(k,2*x))^3. - Karol A. Penson Aug 18 2006
a(n) = Sum_{i=0..n} (-2)^(n-i)*binomial(n, i)*(Sum_{j=0..i} binomial(i, j)^3). - Vasu Tewari (vasu(AT)math.ubc.ca), Aug 04 2010
O.g.f.: (4/Pi)*EllipticK( 8*sqrt(z^3*(1+3*z))/(1-12*z^2+sqrt((1-6*z)*(1+2*z)^3)) ) / sqrt(2 - 24*z^2 + 2*sqrt((1-6*z)*(1+2*z)^3)). - Sergey Perepechko, Feb 08 2011
O.g.f.: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)*(1+2*x)^n. - Paul D. Hanna, Feb 26 2012
a(n) ~ sqrt(3)*6^n/(2*Pi*n). - Vaclav Kotesovec, Aug 13 2013
O.g.f.: 2F1(1/3,2/3; 1; 27*x^2*(1+2*x)). - R. J. Mathar, Sep 29 2020

More terms from David Bloom, Mar 1997

Formula and further terms from Cyril Banderier, Oct 12 2000

A260793 Primes p such that p does not divide any term of the Apéry-like sequence A000172 (also known as Type I primes).

Original entry on oeis.org

3, 11, 17, 19, 43, 83, 89, 97, 113, 137, 139, 163, 193, 211, 233, 241, 283, 307, 313, 331, 347, 353, 379, 401, 409, 419, 433, 443, 491, 499, 523, 547, 569, 587, 601, 617, 619, 641, 643, 673, 811, 827, 859, 881, 929, 947, 953, 977, 1009, 1019, 1033, 1049, 1051

Offset: 1

N. J. A. Sloane, Aug 05 2015

nonn

See Schulte et al. (2014) for the precise definition of Type I primes.

Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5
Amita Malik and Armin Straub, Lists of all primes up to 10000 in A133370 and A260793, A291275-A291284, together with Mathematica code.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5
A. Schulte, S. VanSchalkwyk, A. Yang, On the divisibility and valuations of the Franel numbers, in MSRI-UP Research Reports, 2014.
A. Schulte, S. VanSchalkwyk, A. Yang, On the divisibility and valuations of the Franel numbers, Examples of Outstanding Student Posters, MAA.

Cf. A260791, A260792.

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

maxPrime = 1051;
maxPi = PrimePi @ maxPrime;
okQ[p_] := AllTrue[Range[3 maxPi (* coeff 3 is empirical *)], GCD[HypergeometricPFQ[{-#, -#, -#}, {1, 1}, -1], p] == 1&];
Select[Prime[Range[maxPi]], okQ] (* Jean-François Alcover, Jan 13 2020 *)

Edited by N. J. A. Sloane, Aug 22 2017

A133370 Primes p such that p does not divide any term of the Apery sequence A005259 .

Original entry on oeis.org

2, 3, 7, 13, 23, 29, 43, 47, 53, 67, 71, 79, 83, 89, 101, 103, 107, 109, 113, 127, 131, 137, 149, 157, 167, 173, 199, 223, 229, 239, 263, 269, 277, 281, 311, 313, 317, 337, 349, 353, 359, 373, 383, 389, 397, 401, 409, 421, 449, 457, 461, 467, 479, 487, 491

Offset: 1

Philippe Deléham, Oct 27 2007

nonn

Malik and Straub give arguments suggesting that this sequence is infinite. - N. J. A. Sloane, Aug 06 2017

Robert Price, Table of n, a(n) for n = 1..758
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.
Amita Malik, Mathematica notebook for generating this sequence and A260793, A291275-A291284
Amita Malik, List of all primes up to 10000 in this sequence and in A260793, A291275-A291284, together with Mathematica code.
E. Rowland, R. Yassawi, Automatic congruences for diagonals of rational functions, arXiv preprint arXiv:1310.8635 [math.NT], 2013.

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.

NeverDividesLucasSeqQ[a_, p_] := And @@ Table[Mod[a[n], p]>0, {n, 0, p-1}];
A3[a_, b_, c_, n_ /; n < 0] = 0;
A3[a_, b_, c_, 0] = 1;
A3[a_, b_, c_, n_] := A3[a, b, c, n] = (((2n - 1)(a (n-1)^2 + a (n-1) + b)) A3[a, b, c, n-1] - c (n-1)^3 A3[a, b, c, n-2])/n^3;
A3[a_, b_, c_, d_, n_ /; n < 0] = 0;
Agamma[n_] := A3[17, 5, 1, n];
Select[Range[1000], PrimeQ[#] && NeverDividesLucasSeqQ[Agamma, #]&] (* Jean-François Alcover, Aug 05 2018, copied from Amita Malik's notebook *)

Terms a(16) onwards computed by Amita Malik - N. J. A. Sloane, Aug 21 2017

A291275 Primes p such that p does not divide any term of the Apéry-like sequence A005258.

Original entry on oeis.org

2, 5, 13, 17, 29, 37, 41, 61, 73, 89, 101, 109, 137, 149, 173, 181, 197, 229, 233, 269, 277, 313, 337, 349, 353, 373, 397, 401, 409, 433, 457, 461, 541, 557, 601, 613, 641, 661, 673, 677, 701, 709, 733, 761, 769, 797, 821, 829, 853, 857, 877, 929, 941, 977

Offset: 1

N. J. A. Sloane, Aug 21 2017

nonn

Amita Malik and Armin Straub, Mathematica notebook for generating A133370 and A260793, A291275-A291284
Amita Malik and Armin Straub, Lists of all primes up to 10000 in A133370 and A260793, A291275-A291284, together with Mathematica code.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.

maxPrime = 977;
maxPi = PrimePi @ maxPrime;
okQ[p_] := AllTrue[Range[3 maxPi (* coeff 3 is empirical *)], GCD[HypergeometricPFQ[{# + 1, -#, -#}, {1, 1}, 1], p] == 1&];
Select[Prime[Range[maxPi]], okQ] (* Jean-François Alcover, Jan 13 2020 *)

A291284 Primes p such that p does not divide any term of the Apery-like sequence A290576.

Original entry on oeis.org

2, 5, 7, 13, 17, 19, 29, 37, 43, 47, 59, 61, 67, 71, 83, 89, 101, 109, 127, 139, 149, 167, 173, 191, 211, 233, 239, 241, 251, 257, 271, 277, 281, 307, 311, 313, 331, 337, 347, 349, 353, 359, 373, 379, 383, 409, 419, 421, 431, 433, 443

Offset: 1

N. J. A. Sloane, Aug 21 2017

nonn

Amita Malik and Armin Straub, Mathematica notebook for generating A133370 and A260793, A291275-A291284
Amita Malik and Armin Straub, Lists of all primes up to 10000 in A133370 and A260793, A291275-A291284, together with Mathematica code.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5

A064037 Number of walks of length 2n on cubic lattice, starting and finishing at origin and staying in first (nonnegative) octant.

Original entry on oeis.org

1, 3, 24, 285, 4242, 73206, 1403028, 29082339, 640672890, 14818136190, 356665411440, 8874875097270, 227135946200940, 5955171596514900, 159439898653636320, 4347741997166750235, 120493374240909299130, 3387806231071627372590, 96488484001399878973200

Offset: 0

Henry Bottomley, Aug 23 2001

nonn

			a(1)=3 and a(2)=24 since if the possible steps are Right, Left, Up, Down, Forwards and Backwards, then the two-step paths are FB, RL and UD, while the four-step paths are FBFB, FBRL, FBUD, FFBB, FRBL, FRLB, FUBD, FUDB, RFBL, RFLB, RLFB, RLRL, RLUD, RRLL, RUDL, RULD, UDFB, UDRL, UDUD, UFBD, UFDB, URDL, URLD, UUDD.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
James Mallos, A 6-Letter 'DNA' for Baskets with Handles, Mathematics (2019) Vol. 7, No. 2, 165.
G. Xin, Determinant formulas relating to tableaux of bounded height, Adv. Appl. Math. 45 (2010) 197-211.

Cf. A064036. The two- and one-dimensional equivalents are A005568 and A000108.

f := -3*x+(1+sqrt(1-40*x+144*x^2))/4;
H := (1-2*f)*f*hypergeom([1/6,1/3],[1],27*(1-2*f)*f^2)^2/sqrt(1+6*f);
r2 := (1-4*x)*(36*x-1)*(1920*x^2+166*x+1)*x^2;
r1 := -(138240*x^4+7776*x^3+200*x^2-92*x-1)*x;
r0 := 19800*x^3+764*x^2-86*x-1;
ogf := (r2*diff(H,x,x)+r1*diff(H,x)+r0*H)/(5760*x^4) + 1/(2*x);
series(ogf, x=0, 30); # Mark van Hoeij, Apr 19 2013
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 2*n+1, ((8*n-4)*(5*n^2+10*n+3)
       *a(n-1)-36*(2*n-1)*(2*n-3)*(n-1)*a(n-2))/((n+1)*(n+2)*(n+3)))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 29 2019

Table[Sum[Binomial[2*n, 2*j] * CatalanNumber[j] * CatalanNumber[j+1] * CatalanNumber[n-j], {j, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 09 2019 *)

C(n,k) = binomial(n,k);
c(n) = binomial(2*n,n)/(n+1);
a(n) = sum(j=0,n, C(2*n, 2*j)*c(j)*c(j+1)*c(n-j));
/* Joerg Arndt, Apr 19 2013 */

a(n) = Sum_{j=0..n} C(2n, 2j)*c(j)*c(j+1)*c(n-j) where c(k)=A000108(k).
G.f. is a large expression in terms of hypergeometric functions and sqrt's, see Maple program. - Mark van Hoeij, Apr 19 2013
a(n) = binomial(2*n,n)*((7*n+11)*A002893(n+1)-(9*n+9)*A002893(n))/(2*(n+1)*(n+2)^2*(n+3)). - Mark van Hoeij, Apr 19 2013
a(n) ~ 2^(2*n - 2) * 3^(2*n + 9/2) / (Pi^(3/2) * n^(9/2)). - Vaclav Kotesovec, Jun 09 2019
D-finite with recurrence: (n+3)*(n+2)*(n+1)*a(n) -4*(2*n-1)*(5*n^2+10*n+3)*a(n-1) +36*(n-1)*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Feb 20 2020

Added more terms, Joerg Arndt, Apr 19 2013

A287316 Square array A(n,k) = (n!)^2 [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 15, 20, 1, 0, 1, 5, 28, 93, 70, 1, 0, 1, 6, 45, 256, 639, 252, 1, 0, 1, 7, 66, 545, 2716, 4653, 924, 1, 0, 1, 8, 91, 996, 7885, 31504, 35169, 3432, 1, 0, 1, 9, 120, 1645, 18306, 127905, 387136, 272835, 12870, 1, 0

Offset: 0

Peter Luschny, May 23 2017

A287314 provide polynomials and A287315 rational functions generating the columns of the array.

			Arrays start:
k\n| 0  1    2      3         4        5          6           7
---|----------------------------------------------------------------
k=0| 1, 0,   0,      0,       0,       0,         0,          0, ... A000007
k=1| 1, 1,   1,      1,       1,       1,         1,          1, ... A000012
k=2| 1, 2,   6,     20,      70,     252,       924,       3432, ... A000984
k=3| 1, 3,  15,     93,     639,    4653,     35169,     272835, ... A002893
k=4| 1, 4,  28,    256,    2716,   31504,    387136,    4951552, ... A002895
k=5| 1, 5,  45,    545,    7885,  127905,   2241225,   41467725, ... A169714
k=6| 1, 6,  66,    996,   18306,  384156,   8848236,  218040696, ... A169715
k=7| 1, 7,  91,   1645,   36715,  948157,  27210169,  844691407, ...
k=8| 1, 8, 120,   2528,   66424, 2039808,  70283424, 2643158400, ... A385286
k=9| 1, 9, 153,   3681,  111321, 3965409, 159700401, 7071121017, ...
       A000384,A169711, A169712, A169713,                            A033935

Jeremy Tan, Antidiagonals n = 0..50, flattened
Nikolai Beluhov, Powers of 2 in High-Dimensional Lattice Walks, arXiv:2506.12789 [math.CO], 2025. See p. 19.
Ryan S. Bennink, Counting Abelian Squares for a Problem in Quantum Computing, arXiv:2208.02360 [quant-ph], 2022.
Jonathan M. Borwein, A short walk can be beautiful, preprint, Journal of Humanistic Mathematics, Volume 6 Issue 1 (January 2016), pages 86-109.
Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals, preprint, Theoretical Computer Science, Volume 479, 1 April 2013, Pages 4-21.
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Some Arithmetic Properties of Short Random Walk Integrals, preprint, FPSAC 2010, San Francisco, USA.
L. Bruce Richmond and Jeffrey 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.

Rows: A000007 (k=0), A000012 (k=1), A000984 (k=2), A002893 (k=3), A002895 (k=4), A169714 (k=5), A169715 (k=6), A385286 (k=8).
Columns: A001477(n=1), A000384 (n=2), A169711 (n=3), A169712 (n=4), A169713 (n=5).
Cf. A033935 (diagonal), A287314, A287315, A287318.

A287316_row := proc(k, len) local b, ser;
b := k -> BesselI(0, 2*sqrt(x))^k: ser := series(b(k), x, len);
seq((i!)^2*coeff(ser,x,i), i=0..len-1) end:
for k from 0 to 6 do A287316_row(k, 9) od;
A287316_col := proc(n, len) local k, x;
sum(z^k/k!^2, k = 0..infinity); series(%^x, z=0, n+1):
unapply(n!^2*coeff(%, z, n), x); seq(%(j), j=0..len) end:
for n from 0 to 7 do A287316_col(n, 9) od;

Table[Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^k, {x, 0, n}] (n!)^2, {n, 0, 6}], {k, 0,9}]

A287316_row(K, N) = {
  my(x='x + O('x^(2*N-1)));
  Vec(serlaplace(serlaplace(substpol(besseli(0,2*x)^K, 'x^2, 'x))));
};
N=8; concat([vector(N, n, n==1)], vector(9, k, A287316_row(k, N))) \\ Gheorghe Coserea, Jan 12 2018

{A(n, k) = if(n<0 || k<0, 0, n!^2 * polcoeff(besseli(0, 2*x + x*O(x^(2*n)))^k, 2*n))}; /* Michael Somos, Dec 30 2021 */

A(k, n) = my(x='x+O('x^(n+1))); n!^2*polcoeff(hypergeom([], [1], x)^k, n); \\ Peter Luschny, Jun 24 2025

from math import comb
from functools import lru_cache
@lru_cache(maxsize=None)
def A(n,k):
    if k <= 0: return 0**n
    return sum(A(i,k-1)*comb(n,i)**2 for i in range(n+1))
for k in range(10): print([A(n, k) for n in range(8)])
# Jeremy Tan, Dec 10 2021

A(n,k) = A287318(n,k) / binomial(2*n,n).

If a+b=k then A(n,k) = Sum_{i=0..n} A(i,a)*A(n-i,b)*binomial(n,i)^2 (Richmond and Shallit). In particular A(n,k) = Sum_{i=0..n} A(i,k-1)*binomial(n,i)^2. - Jeremy Tan, Dec 10 2021

A108304 Number of set partitions of {1, ..., n} that avoid 3-crossings.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 202, 859, 3930, 19095, 97566, 520257, 2877834, 16434105, 96505490, 580864901, 3573876308, 22426075431, 143242527870, 929759705415, 6123822269373, 40877248201308, 276229252359846, 1887840181793185, 13037523684646810, 90913254352507057

Offset: 0

Mireille Bousquet-Mélou, Jun 29 2005

There is also a sum-formula for a(n). See Bousquet-Mélou and Xin.

Also partitions avoiding a certain pattern (see J. Bloom and S. Elizalde). - N. J. A. Sloane, Jan 02 2013

			There are 203 partitions of 6 elements, but a(6)=202 because the partition (1,4)(2,5)(3,6) has a 3-crossing.
G.f. = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 202*x^6 + 859*x^7 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Jonathan Bloom and Sergi Elizalde, Pattern avoidance in matchings and partitions, arXiv preprint arXiv:1211.3442 [math.CO], 2012.
Alin Bostan, Jordan Tirrell, Bruce W. Westbury, Yi Zhang, On sequences associated to the invariant theory of rank two simple Lie algebras, arXiv:1911.10288 [math.CO], 2019.
Alin Bostan, Jordan Tirrell, Bruce W. Westbury and Yi Zhang, On some combinatorial sequences associated to invariant theory, arXiv:2110.13753 [math.CO], 2021.
M. Bousquet-Mélou and G. Xin, On partitions avoiding 3-crossings, arXiv:math/0506551 [math.CO], 2005-2006.
Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615 [math.CO], 2011.
Wei Chen, Enumeration of Set Partitions Refined by Crossing and Nesting Numbers, MS Thesis, Department of Mathematics. Simon Fraser University, Fall 2014.
W. Chen, E. Deng, R. Du, R. Stanley, and C. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005.
Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, Pattern-avoiding ascent sequences of length 3, arXiv:2111.01279 [math.CO], Nov 01 2021.
P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641 [math.CO], 2011.
Juan B. Gil, Jordan O. Tirrell, A simple bijection for classical and enhanced k-noncrossing partitions, arXiv:1806.09065 [math.CO], 2018.
Zhicong Lin, Restricted inversion sequences and enhanced 3-noncrossing partitions, arXiv:1706.07213 [math.CO], 2017.
Zhicong Lin, Sherry H. F. Yan, Vincular patterns in inversion sequences, Applied Mathematics and Computation (2020), Vol. 364, 124672.
T. Mansour and M. Shattuck, Some enumerative results related to ascent sequences, arXiv preprint arXiv:1207.3755 [math.CO], 2012. - _N. J. A. Sloane_, Dec 22 2012
Marni Mishna and Lily Yen, Set partitions with no k-nesting, arXiv preprint arXiv:1106.5036 [math.CO], 2011.

a[0] = a[1] = 1; a[n_] := a[n] = (2*(5*n^2 + 12*n - 2)*a[n-1] + 9*(-n^2 + n + 2)*a[n-2])/((n+4)*(n+5)); Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 30 2015 *)

v = vector(66,n,n);
for (n=1, #v-2, v[n+2] = ((10*n^2+64*n+84)*v[n+1]-(9*n^2+27*n)*v[n]) / (n^2+13*n+42) );
vector(#v+1,n, if(n==1,1,v[n-1])) \\ Joerg Arndt, Sep 01 2012

Recurrence: (9*n^2+27*n) * a(n) + (-10*n^2-64*n-84) * a(n+1) + (n^2+13*n+42) * a(n+2) = 0.
a(n) = (-18*(n+1)*(4*n^5+73*n^4+530*n^3+1928*n^2+3654*n+2916)*A002893(n)+(8*n^6+17156*n^2+6084*n^3+17496+27612*n+1358*n^4+162*n^5) *A002893(n+1))/ (3*n*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)). - Mark van Hoeij, Nov 05 2011
G.f.: (1+7*x-20*x^2+30*x^3-18*x^4-(3*x+1)^2*(x-1)^2*hypergeom([-2/3, -1/3],[2],27*x*(x-1)^2/(3*x+1)^3))/(6*x^4). - Mark van Hoeij, Nov 05 2011
a(n) ~ 5 * sqrt(3) * 3^(2*n+9) / (32*Pi*n^7), Bousquet-Mélou and Xin, 2006. - Vaclav Kotesovec, Aug 23 2014

More terms added by Joerg Arndt, Sep 01 2012

cp's OEIS Frontend

A262177 Decimal expansion of Q_5 = zeta(5) / (Sum_{k>=1} (-1)^(k+1) / (k^5 * binomial(2k, k))), a conjecturally irrational constant defined by an Apéry-like formula.

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

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A002898 Number of n-step closed paths on hexagonal lattice.

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A260793 Primes p such that p does not divide any term of the Apéry-like sequence A000172 (also known as Type I primes).

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A133370 Primes p such that p does not divide any term of the Apery sequence A005259 .

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A291275 Primes p such that p does not divide any term of the Apéry-like sequence A005258.

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A291284 Primes p such that p does not divide any term of the Apery-like sequence A290576.

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A064037 Number of walks of length 2n on cubic lattice, starting and finishing at origin and staying in first (nonnegative) octant.

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