A287318 -id:A287318 - cp's OEIS frontend

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

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

A152746 Six times hexagonal numbers: 6n(2*n-1).

Original entry on oeis.org

0, 6, 36, 90, 168, 270, 396, 546, 720, 918, 1140, 1386, 1656, 1950, 2268, 2610, 2976, 3366, 3780, 4218, 4680, 5166, 5676, 6210, 6768, 7350, 7956, 8586, 9240, 9918, 10620, 11346, 12096, 12870, 13668, 14490, 15336, 16206, 17100

Offset: 0

Omar E. Pol, Dec 12 2008

Sequence found by reading the line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Sep 18 2011

a(n) is the number of walks on a cubic lattice of n dimensions that return to the origin, not necessarily for the first time, after 4 steps. - Shel Kaphan, Mar 20 2023

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000384, A001082, A002939, A094159, A085250, A152745.

Column n=2 of A287318.

[6*n*(2*n-1): n in [0..50]]; // G. C. Greubel, Sep 01 2018

6*PolygonalNumber[6,Range[0,40]] (* The program uses the PolygonalNumber function from Mathematica version 10 *) (* Harvey P. Dale, Mar 04 2016 *)
LinearRecurrence[{3,-3,1}, {0,6,36}, 50] (* or *) Table[6*n*(2*n-1), {n,0,50}] (* G. C. Greubel, Sep 01 2018 *)

a(n)=6*n*(2*n-1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 12*n^2 - 6*n = A000384(n)*6 = A002939(n)*3 = A094159(n)*2.
a(n) = a(n-1) + 24*n - 18 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
From G. C. Greubel, Sep 01 2018: (Start)
G.f.: 6*x*(1+3*x)/(1-x)^3.
E.g.f.: 6*x*(1+2*x)*exp(x). (End)
From Amiram Eldar, Mar 30 2023: (Start)
Sum_{n>=1} 1/a(n) = log(2)/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/12 - log(2)/6. (End)

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

A039699 Number of 4-dimensional cubic lattice walks that start and end at the origin after 2n steps, free to pass through origin at intermediate stages.

Original entry on oeis.org

1, 8, 168, 5120, 190120, 7939008, 357713664, 16993726464, 839358285480, 42714450658880, 2225741588095168, 118227198981126144, 6380762273973278464, 349019710593278412800, 19310744204362333900800, 1079054103459778710405120, 60818479243449308702049960

Offset: 0

Alessandro Zinani (alzinani(AT)tin.it)

Generating function G(x) is D-finite with a singular point at x = 1/64 (cf. Graph Link). After summing 300000 terms, G(1/64) = 1.239466... and 1 - 1/G(1/64) = 0.193201... Convergence to A086232 is very slow. - Bradley Klee, Aug 20 2018

a(n) is also the constant term in the expansion of (w + 1/w + x + 1/x + y + 1/y + z + 1/z)^(2n). This follows directly from the sequence name, each variable corresponding to a single step in one of the four axis directions. - Christopher J. Smyth, Sep 28 2018

			a(5)=7939008, i.e., there are 7939008 different walks that start and end at origin of a 4-dimensional integer lattice after 2*5=10 steps, free to pass through origin at intermediate steps.

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.

Seiichi Manyama, Table of n, a(n) for n = 0..557
S. R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice. [broken link]
Steven R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice. [Cached copy, with permission of the author]
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 pp. 33, 44.
Bradley Klee, Graph of g.f.
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, 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.
J. Novak, Pólya's random walk theorem, arXiv:1301.3916 [math.PR], 2013.

1-dimensional, 2-dimensional, 3-dimensional analogs are A000984, A002894, A002896. Pólya Constant: A086232.

Row k=4 of A287318.

A039699 := n -> binomial(2*n,n)^2*hypergeom([1/2, -n, -n, -n],[1, 1, 1/2 - n], 1):
seq(simplify(A039699(n)), n=0..14); # Peter Luschny, May 23 2017

max = 30 (* must be even *); Partition[ CoefficientList[ Series[ BesselI[0, 2 x]^4, {x, 0, max}], x]*Range[0, max]!, 2][[All, 1]] (* Jean-François Alcover, Oct 05 2011 *)
With[{nn=30},Take[CoefficientList[Series[BesselI[0,2x]^4,{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Aug 09 2013 *)
RecurrenceTable[{256*(n-1)^2*(2*n-3)*(2*n-1)*a[n-2] - 4*(2*n-1)^2*(5*n^2-5*n+2)*a[n-1] + n^4*a[n]==0, a[0]==1, a[1]==8}, a, {n,0,100}] (* Bradley Klee, Aug 20 2018 *)

C=binomial;
A002895(n) = sum(k=0,n, C(n,k)^2 * C(2*n-2*k,n-k) * C(2*k,k) );
a(n)= C(2*n,n) * A002895(n);
/* Joerg Arndt, Apr 19 2013 */

from math import comb
def A039699(n): return comb(n<<1,n)*((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

E.g.f.: Sum_{n>=0} a(2*n) * x^(2*n)/(2*n)! = I_0(2*x)^4. (I = Modified Bessel function of the first kind).
a(n) = binomial(2*n,n)*A002895(n). - Mark van Hoeij, Apr 19 2013
a(n) = binomial(2*n,n)^2*hypergeom([1/2,-n,-n,-n],[1,1,1/2-n],1). - Peter Luschny, May 23 2017
a(n) ~ 2^(6*n+1) / (Pi*n)^2. - Vaclav Kotesovec, Nov 13 2017
From Bradley Klee, Aug 20 2018: (Start)
G.f.: Define G(x) = Sum_{n>=0} a(n)*x^n and G^(j) = (d/dx)^j G(x), then Sum_{j=0..4,k=0..5} M_{j,k}*G^(j)*x^k = 0, with
M={{-8, 768, 0, 0, 0, 0}, {1, -424, 14592, 0, 0, 0}, {0, 7, -1172, 25344, 0, 0}, {0, 0, 6, -640, 10240, 0}, {0, 0, 0, 1, -80, 1024}}.
Sum_{j=0..2,k=0..4} M_{j,k}*a(n-j)*n^k = 0, with
M={{0, 0, 0, 0, 1}, {-8, 52, -132, 160, -80}, {768, -3584, 5888, -4096, 1024}}.
(End)
a(n) = Sum_{i+j+k+l=n, 0<=i,j,k,l<=n} multinomial(2n [i,i,j,j,k,k,l,l]). - Shel Kaphan, Jan 16 2023

A361397 Number A(n,k) of k-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 20, 4, 0, 1, 8, 54, 176, 10, 0, 1, 10, 104, 996, 1876, 28, 0, 1, 12, 170, 2944, 22734, 22064, 84, 0, 1, 14, 252, 6500, 108136, 577692, 275568, 264, 0, 1, 16, 350, 12144, 332050, 4525888, 15680628, 3584064, 858, 0

Offset: 0

Alois P. Heinz, Mar 10 2023

Column k is INVERTi transform of k-th row of A287318.

			Square array A(n,k) begins:
  1,  1,     1,      1,       1,        1,        1, ...
  0,  2,     4,      6,       8,       10,       12, ...
  0,  2,    20,     54,     104,      170,      252, ...
  0,  4,   176,    996,    2944,     6500,    12144, ...
  0, 10,  1876,  22734,  108136,   332050,   796860, ...
  0, 28, 22064, 577692, 4525888, 19784060, 62039088, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-5 give: A000007, |A002420|, A054474, A049037, A359801, A361364.
Rows n=0-2 give: A000012, A005843, A139271.
Main diagonal gives A361297.
Cf. A000108, A287318.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
    end:
g:= proc(n, k) option remember; `if` (n<1, -1,
      -add(g(n-i, k)*(2*i)!*b(i, k)/i!^2, i=1..n))
    end:
A:= (n,k)-> `if`(n=0, 1, `if`(k=0, 0, g(n, k))):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, 0] = 0; b[n_, 1] = 1; b[0, k_] = 1;
b[n_, k_] := b[n, k] = Sum[Binomial[n, i]^2*b[i, k - 1], {i, 0, n}]; (* A287316 *)
g[n_, k_] := g[n, k] = b[n, k]*Binomial[2 n, n]; (* A287318 *)
a[n_, k_] := a[n, k] = g[n, k] - Sum[a[i, k]*g[n - i, k], {i, 1, n - 1}];
TableForm[Table[a[n, k], {k, 0, 10}, {n, 0, 10}]] (* Shel Kaphan, Mar 14 2023 *)

A(n,1)/2 = A000108(n-1) for n >= 1.

G.f. of column k: 2 - 1/Integral_{t=0..oo} exp(-t)*BesselI(0,2*t*sqrt(x))^k dt. - Shel Kaphan, Mar 19 2023

A287317 Number of 5-dimensional cubic lattice walks that start and end at origin after 2n steps, free to pass through origin at intermediate stages.

Original entry on oeis.org

1, 10, 270, 10900, 551950, 32232060, 2070891900, 142317232200, 10277494548750, 770878551371500, 59577647564312020, 4717432065143561400, 381091087190569291900, 31308955091335405435000, 2609450031306515140215000, 220199552765301571338488400

Offset: 0

Peter Luschny, May 23 2017

Case k=5 of A287318.
Cf. A169714, A287316
1-4 dimensional analogs are A000984, A002894, A002896, A039699.

A287317_list := proc(len) series(BesselI(0, 2*sqrt(x))^5, x, len);
seq((2*i)!*coeff(%, x, i), i=0..len-1) end: A287317_list(16);

Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^5, {x, 0, n}] (2 n) !, {n, 0, 15}]
Table[Binomial[2n,n]^2 Sum[(Binomial[n,j]^4/Binomial[2n,2j]) HypergeometricPFQ[{-j,-j,-j}, {1,1/2-j}, 1/4], {j,0,n}], {n,0,15}]
Table[Sum[(2 n)!/(i! j! k! l! (n-i-j-k-l)!)^2, {i,0,n}, {j,0,n-i}, {k,0,n-i-j}, {l,0,n-i-j-k}], {n,0,30}] (* Shel Kaphan, Jan 24 2023 *)

a(n) = (2*n)! [x^n] BesselI(0, 2*sqrt(x))^5.
a(n) = binomial(2*n,n)*A169714(n).
a(n) ~ 2^(2*n) * 5^(2*n + 5/2) / (16 * Pi^(5/2) * n^(5/2)). - Vaclav Kotesovec, Nov 13 2017
a(n) = Sum_{i+j+k+l+m=n, 0<=i,j,k,l,m<=n} multinomial(2n, [i,i,j,j,k,k,l,l,m,m]). - Shel Kaphan, Jan 24 2023

Moved original definition to formula section and reworded definition descriptively similar to sequence A039699, by Dave R.M. Langers, Oct 12 2022

A303503 a(n) = (2n)! [x^(2n)] BesselI(0,2x)^n.

Original entry on oeis.org

1, 2, 36, 1860, 190120, 32232060, 8175770064, 2898980908824, 1369263687414480, 830988068906518380, 630109741730668410640, 583773362067938664133512, 648851848280206013365243776, 852146184628067383511375555000, 1305460597778526044143501996708800, 2307324514460203126471248458864413200

Offset: 0

Ilya Gutkovskiy, Apr 25 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..212

Main diagonal of A287318.

Cf. A000984, A002894, A002896, A033935, A039699, A287317, A361297.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
    end:
a:= n-> (2*n)!*b(n$2)/n!^2:
seq(a(n), n=0..17);  # Alois P. Heinz, Jan 29 2023

Table[(2 n)! SeriesCoefficient[BesselI[0, 2 x]^n, {x, 0, 2 n}], {n, 0, 15}]

a(n) = A287318(n,n).
a(n) ~ c * d^n * n^(2*n), where c = 1.72802011936236389522137050964080... and d = 1.1381284656425793765251319541847869000364101065484286935... - Vaclav Kotesovec, Apr 26 2018
a(n) = A000984(n)*A033935(n). - Alois P. Heinz, Jan 30 2023

A356258 Number of 6-dimensional cubic lattice walks that start and end at origin after 2n steps, free to pass through origin at intermediate stages.

Original entry on oeis.org

1, 12, 396, 19920, 1281420, 96807312, 8175770064, 748315668672, 72729762868620, 7402621930738320, 781429888276676496, 84955810313787521472, 9463540456205136873936, 1075903653146632508721600, 124461755084172965028753600, 14615050011682746903615601920

Offset: 0

Dave R.M. Langers, Oct 12 2022

			a(1)=12, because twelve paths start at the origin, visit one of the adjacent vertices, and immediately return to the origin, resulting in 12 different paths of length 2n=2*1=2.

Alois P. Heinz, Table of n, a(n) for n = 0..467

Row k=6 of A287318.

1-5 dimensional analogs are A000984, A002894, A002896, A039699, A287317.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
    end:
a:= n-> (2*n)!*b(n, 6)/n!^2:
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 30 2023

E.g.f.: Sum_{n>=0} a(2*n) * x^(2*n)/(2*n)! = I_0(2*x)^6. (I = Modified Bessel function first kind).

a(n) = Sum_{h+i+j+k+l+m=n, 0<=h,i,j,k,l,m<=n} multinomial(2n [h,h,i,i,j,j,k,k,l,l,m,m]). - Shel Kaphan, Jan 29 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

A152746 Six times hexagonal numbers: 6*n*(2*n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Python

Formula

A039699 Number of 4-dimensional cubic lattice walks that start and end at the origin after 2n steps, free to pass through origin at intermediate stages.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A361397 Number A(n,k) of k-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A287317 Number of 5-dimensional cubic lattice walks that start and end at origin after 2n steps, free to pass through origin at intermediate stages.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A152746 Six times hexagonal numbers: 6n(2*n-1).

A303503 a(n) = (2n)! [x^(2n)] BesselI(0,2x)^n.