A054474 -id:A054474 - cp's OEIS frontend

A002894 a(n) = binomial(2n, n)^2.

1, 4, 36, 400, 4900, 63504, 853776, 11778624, 165636900, 2363904400, 34134779536, 497634306624, 7312459672336, 108172480360000, 1609341595560000, 24061445010950400, 361297635242552100, 5445717990022688400, 82358080713306090000, 1249287673091590440000

Offset: 0

N. J. A. Sloane

a(n) is the number of monotonic paths (only moving N and E) in the lattice [0..2n] X [0..2n] that contain the points (0,0), (n,n) and (2n,2n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004
Expansion of K(k) / (Pi/2) in powers of m/16 = (k/4)^2, where K(k) is the complete elliptic integral of the first kind evaluated at k. - Michael Somos, Mar 04 2003
Square lattice walks that start and end at origin after 2n steps. - Gareth McCaughan and Michael Somos, Jun 12 2004
If A is a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic) then a(n)=E[(tr(A^k))^{2n}] for any k > 4. - Andrew V. Sutherland, Apr 01 2008
From R. H. Hardin, Feb 03 2016 and R. J. Mathar, Feb 18 2016: (Start)
Also, number of 2 X (2n) arrays of permutations of 2n copies of 0 or 1 with row sums equal.
For example, some solutions for n=3:
  0 1 0 1 0 1   0 1 0 1 1 0   0 0 1 0 1 1   1 1 1 0 0 0
  1 0 0 0 1 1   1 1 0 1 0 0   0 0 0 1 1 1   0 0 1 1 0 1
There is a simple combinatorial argument to show that this is a(n): We have 2n copies of 0's and 1's and need equal row sums. Therefore there must be n 1's in each of the two rows. Otherwise there are no constraints, so there are C(2n,n) ways of placing the 1's in the first row and independently C(2n,n) ways of placing the 1's in the second. The product is clearly C(2n,n)^2. (End)
Also the even part of the bisection of A241530. One half of the odd part is given in A000894. - Wolfdieter Lang, Sep 06 2016
From Peter Bala, Jan 26 2018: (Start)
Let S = {[1,0,0], [0,1,0], [1,0,1], [0,1,1]} be a set of four column vectors. Then a(n) equals the number of 3 X k arrays whose columns belong to the set S and whose row sums are all equal to n (apply Eger, Theorem 3). An example is given below. Equivalently, a(n) equals the number of lattice paths from (0,0,0) to (n,n,n) using steps (1,0,0), (0,1,0), (1,0,1) and (0,1,1).
The o.g.f. for the sequence equals the diagonal of the rational function 1/(1 - (x + y + x*z + y*z)).
Row sums of A069466. (End)
Also, the constant term in the expansion of (x + 1/x + y + 1/y)^(2n). - Christopher J. Smyth, Sep 26 2018
Number of ways to place 2n^2 nonattacking pawns on a 2n x 2n board. - Tricia Muldoon Brown, Dec 12 2018
For n>0, a(n) is the number of Littlewood polynomials of degree 4n-1 that have a closed Lill path. A polynomial p(x) has a closed Lill path if and only if p(x) is divisible by x^(2)+1. - Raul Prisacariu, Aug 28 2024

			G.f. = 1 + 4*x + 36*x^2 + 400*x^3 + 4900*x^4 + 63504*x^5 + 853776*x^6 + ... - _Michael Somos_, Aug 06 2014
From _Peter Bala_, Jan 26 2018: (Start)
a(2) = 36: The thirty six 3 x k arrays with columns belonging to the set of column vectors S = {[1,0,0], [0,1,0], [1,0,1], [0,1,1]} and having all row sums equal to 2 are the 6 distinct arrays obtained by permuting the columns of
  /1 1 0 0\
  |0 0 1 1|,
  \0 0 1 1/
the 6 distinct arrays obtained by permuting the columns of
  /0 0 1 1\
  |1 1 0 0|
  \0 0 1 1/
and the 24 arrays obtained by permuting the columns of
  /1 0 1 0\
  |0 1 0 1|. (End)
  \0 0 1 1/

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 591,828.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
Leonard Lipshitz and A. van der Poorten. "Rational functions, diagonals, automata and arithmetic." In Number Theory, Richard A. Mollin, ed., Walter de Gruyter, Berlin (1990): 339-358.
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).

T. D. Noe, Table of n, a(n) for n = 0..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
R. Bacher, Meander algebras, Institut Fourier, 1999.
E. Barcucci, A. Frosini and S. Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005). 62-78.
Arnaud Beauville, Les familles stables de courbes elliptiques sur P^1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982, 657-660. MR0664643 (83h:14008)
Alin Bostan, Armin Straub, and Sergey Yurkevich, On the representability of sequences as constant terms, arXiv:2212.10116 [math.NT], 2022.
Tricia Muldoon Brown, The Problem of Pawns, The Electronic Journal of Combinatorics (2019) Vol. 26, Issue 3, #P3.21. Also arXiv:1811.09606, [math.CO], 2018.
John Maxwell Campbell, New series involving harmonic numbers and squared central binomial coefficients, Rocky Mountain J. Math., 49 (2019), 2513-2544.
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
Murray Elder, Cogrowth, 2011.
M. Elder, A. Rechnitzer, E. J. Janse van Rensburg, and T. Wong, The cogrowth series for BS(N,N) is D-finite, arXiv:1309.4184 [math.GR], 2013.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 90.
Florian Fürnsinn and Sergey Yurkevich, Algebraicity of hypergeometric functions with arbitrary parameters, arXiv:2308.12855 [math.CA], 2023.
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.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2010.
Markus Kuba and Alois Panholzer, Lattice paths and the diagonal of the cube, arXiv:2411.03930 [math.CO], 2024. See p. 14.
L. Lipshitz and A. J. van der Poorten, Rational functions, diagonals, automata and arithmetic
Raul Prisacariu, Littlewood Polynomials of Degree n with Closed Lill Paths
Eric M. Rains, High powers of random elements of compact Lie groups, Probability Theory and Related Fields 107 (1997), 219-241.
Grzegorz Siudem and Agata Fronczak, Bell polynomials in the series expansions of the Ising model, arXiv:2007.16132 [math-ph], 2020.
Eric Weisstein's World of Mathematics, Lattice Path.
D. Zagier, Integral solutions of Apéry-like recurrence equations. See line G in sporadic solutions table of page 5.

Row sums of A069466.
Row 2 of A268367 (even terms).
Equals 4*A060150.
Cf. A000984, A000515, A010370, A054474 (INVERTi transform), A172390, A000897, A002897, A006480, A008977, A186420, A188662, A000894, A241530, A002898 (walks hex lattice).

[Binomial(2*n, n)^2: n in [0..20]]; // Vincenzo Librandi, Aug 07 2014

Maple
```
A002894 := n-> binomial(2*n,n)^2.
```

CoefficientList[Series[Hypergeometric2F1[1/2, 1/2, 1, 16x], {x, 0, 20}], x]
Table[Binomial[2n,n]^2,{n,0,20}] (* Harvey P. Dale, Jul 06 2011 *)
a[ n_] := SeriesCoefficient[ EllipticK[16 x] / (Pi/2), {x, 0, n}]; (* Michael Somos, Aug 06 2014 *)
a[n_] := 16^n HypergeometricPFQ[{1/2, -2 n, 2 n + 1}, {1, 1}, 1];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Mar 14 2018 *)

PARI
```
{a(n) = binomial(2*n, n)^2};
```

{a(n) = if( n<0, 0, polcoeff( polcoeff( polcoeff( 1 / (1 - x * (y + z + 1/y + 1/z)) + x * O(x^(2*n)), 2*n), 0), 0))}; /* Michael Somos, Jun 12 2004 */

[binomial(2*n, n)**2 for n in range(17)]  # Zerinvary Lajos, Apr 21 2009

D-finite with recurrence: (n+1)^2*a(n+1) = 4*(2*n + 1)^2*a(n). - Matthijs Coster, Apr 28 2004
a(n) ~ Pi^(-1)*n^(-1)*2^(4*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
G.f.: F(1/2, 1/2; 1; 16*x) = 1 / AGM(1, (1 - 16*x)^(1/2)) = K(4*sqrt(x)) / (Pi/2), where AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre. - Michael Somos, Mar 04 2003
G.f.: 2*EllipticK(4*sqrt(x))/Pi, using Maple's convention for elliptic integrals.
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2x)^2.
a(n) = A000984(n)^2. - Jonathan Vos Post, Jun 17 2007
E.g.f.: (BesselI(0, 2*x))^2 = 1+2*x^2/(U(0)-2*x^2); U(k) = 2*x^2*(2*k+1)+(k+1)^3-2*x^2*(2*k+3)*(k+1)^3/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 23 2011
In generally, for (BesselI(b, 2x))^2=((x^(2*b))/(GAMMA(b+1))^2)*(1+(2*x^2)*(2*b+1)/(Q(0)-(2*x^2)*(2*b+1)); Q(k)=(2*x^2)*(2*k+2*b+1)+(k+1)*(k+b+1)*(k+2*b+1)-(2*x^2)*(k+1)*(k+b+1)*(k+2*b+1)*(2*k+2*b+3)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 23 2011
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 4*(2*k+1)^2*x*(1+4*x)^2/(4*(2*k+1)^2*x*(1+4*x)^2 + (k+1)^2*(1+4*x)^2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 01 2013
0 = +a(n)*(+393216*a(n+2) -119040*a(n+3) +6860*a(n+4)) +a(n+1)*(-16128*a(n+2) +6928*a(n+3) -465*a(n+4)) +a(n+2)*(+36*a(n+2) -63*a(n+3) +6*a(n+4)) for all n in Z. - Michael Somos, Aug 06 2014
Integral representation as the n-th moment of a positive function W(x) on (0,16), in Maple notation, W(x) = EllipticK(sqrt(1-x/16))/(2*Pi^2*sqrt(x)); a(n) = Integral_{x=0..16} x^n*W(x) dx, n>=0. The function W(x) is singular at x=0 and W(16) = 1/(16*Pi). This representation is unique since W(x) is the solution of the Hausdorff moment problem. - Stanley Smith and Karol A. Penson, Jun 19 2015
a(n) ~ 16^n*(2-2/(8*n+2)^2+21/(8*n+2)^4-671/(8*n+2)^6+45081/(8*n+2)^8)^2/((4*n+1)* Pi). - Peter Luschny, Oct 14 2015
a(n) = binomial(2*n,n)*binomial(2*n,n) = ( [x^n](1 + x)^(2*n) ) *( [x^n](1 + x)^(2*n) ) = [x^n](F(x)^(4*n)), where F(x) = 1 + x + x^2 + 4*x^3 + 20*x^4 + 120*x^5 + 798*x^6 + 5697*x^7 + ... appears to have integer coefficients. For similar results see A000897, A002897, A006480, A008977, A186420 and A188662. - Peter Bala, Jul 14 2016
a(n) = Sum_{k = 0..n} binomial(2*n + k,k)*binomial(n,k)^2. Cf. A005258(n) = Sum_{k = 0..n} binomial(n + k,k)*binomial(n,k)^2. - Peter Bala, Jul 27 2016
a(n) = A241530(2*n), n >= 0. - Wolfdieter Lang, Sep 06 2016
E.g.f.: 2F2(1/2,1/2; 1,1; 16*x). - Ilya Gutkovskiy, Jan 23 2018
a(n) = 16^n*hypergeom([1/2, -2*n, 2*n + 1], [1, 1], 1). - Peter Luschny, Mar 14 2018
The right-hand side of the binomial coefficient identity Sum_{k = 0..n} C(n,k)*C(n+k,k)*C(2*n+2*k,n+k)*(-4)^(n-k) = a(n). - Peter Bala, Mar 16 2018
a(n) = [x^n] (1 - x)^(2*n) * P(2*n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. Compare with A245086(n) = [x^n] (1 - x)^(2*n) * P(n,(1 + x)/(1 - x)). - Peter Bala, Mar 23 2022
a(n) = Sum_{k=0..n} multinomial(2n [k k (n-k) (n-k)]), which is another way to count random walks on Z^2, with steps of (0,+-1) or (+-1,0), that return to the point of origin after 2n steps (not necessarily for the first time), as is C(2n,n)^2. - Shel Kaphan, Jan 12 2023
0 = a(n)*(+393216*a(n+2) -119040*a(n+3) +6860*a(n+4)) +a(n+1)*(-16128*a(n+2) +6928*a(n+3) -465*a(n+4)) +a(n+2)*(+36*a(n+2) -63*a(n+3) +6*a(n+4)) for n>=0. - Michael Somos, May 30 2023
From Peter Bala, Sep 12 2023: (Start)
Right-hand side of the binomial coefficient identities
1) Sum_{k = 0..n} (-1)^(n+k) * C(n,k)*C(n+k,n)*C(2*n+k,n) = a(n).
2) 2*Sum_{k = 0..n} (-1)^(n+k) * C(n,k)*C(n+k-1,n)*C(2*n+k-1,n) = a(n) for n >= 1.
3) (4/3)*Sum_{k = 0..n} (-1)^(n+k) * C(n,k)*C(n+k,n)*C(2*n+k-1,n) = a(n) for n >= 1. (End)

Edited by N. J. A. Sloane, Feb 18 2016

A094061 Number of n-moves paths of a king starting and ending at the origin of an infinite chessboard.

Original entry on oeis.org

1, 0, 8, 24, 216, 1200, 8840, 58800, 423640, 3000480, 21824208, 158964960, 1171230984, 8668531872, 64574844048, 483114856224, 3630440899800, 27379154692032, 207172490054816, 1572194644061184, 11962847247681616, 91242602561647680, 697438669619791008

Offset: 0

Matthijs Coster, Apr 29 2004

The chessboard here is the full four-quadrant board Z X Z.
This is an analog of A054474 for walks on a square grid where the steps can be made diagonally as well.
a(n) is the constant term in the expansion of ((x + 1/x) * (y + 1/y) + x^2 + 1/x^2 + y^2 + 1/y^2)^n. - Seiichi Manyama, Nov 03 2019

D. Joyner, "Adventures in Group Theory: Rubik's Cube, Merlin's Machine and Other Mathematical Toys", Johns Hopkins University Press, 2002, pp. 79

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Peter Bala, A note on A094061

Row 2 of A327751.

Cf. A002426, A098070, A126869, A253974, A254129, A254459, A329024.

a:=array(0..30):a[0]:=1:a[1]:=0:a[2]:=8:a[3]:=24:for n from 3 to 29 do a[n+1]:= (n*(5*n+1)*a[n]+2*(15*n^2+6*n-5)*a[n-1]-8*(5*n^2-23*n+21)*a[n-2]-64*(n-2)^2*a[n-3])/(n+1)^2: print(n+1,a[n+1]) od:
# second Maple program
a:= proc(n) option remember; `if`(n<3, (n-1)*(9*n-2)/2,
      ((n-1)*(3*n-1)*(3*n-4) *a(n-1)
      +(108*n^3-396*n^2+452*n-152) *a(n-2)
      +32*(3*n-2)*(n-2)^2 *a(n-3))/ (n^2*(3*n-5)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 02 2012

a[n_]:=Module[{f=(x+x^-1+y+y^-1+x y+x^-1y+x^-1y^-1+x y^-1)^n,s}, s=Series[f,{x,0,0},{y,0,0}]; SeriesCoefficient[s,{0,0}]]; Table[a[n], {n,1,22}] (* Armin Vollmer (Armin.Vollmer(AT)kabelleipzig.de), May 01 2006 *)
CoefficientList[Series[1/(1+4*x)*LegendreP[-1/2,1-32*x*(1+x)/(1+4*x)^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 16 2013 *)

a[0]:1$
a[1]:0$
a[2]:8$
a[3]:24$
a[n]:=((n-1)*(3*n-1)*(3*n-4) *a[n-1]
      +(108*n^3-396*n^2+452*n-152) *a[n-2]
      +32*(3*n-2)*(n-2)^2 *a[n-3])/ (n^2*(3*n-5))$
A094061(n):=a[n]$
makelist(A094061(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

{a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef((1+x+1/x)^k, 0)^2)} \\ Seiichi Manyama, Oct 29 2019

{a(n) = polcoef(polcoef(((x+1/x)*(y+1/y)+x^2+1/x^2+y^2+1/y^2)^n, 0), 0)} \\ Seiichi Manyama, Nov 03 2019

D-finite with recurrence (n+1)^2*a(n+1) = n*(5*n+1)*a(n) + 2*(15*n^2+6*n-5)*a(n-1) - 8*(5*n^2-23*n+21) *a(n-2) - 64*(n-2)^2*a(n-3).
G.f.: (2/(Pi*(1+4*x))) * EllipticK(4*sqrt(x*(1+x))/(1+4*x)) = 1/(1+4*x) * hypergeom([1/2,1/2], [1], 16*(x*(1+x))/(1+4*x)^2). - Sergey Perepechko, Jan 15 2011
a(n) ~ 2^(3*n+1)/(3*Pi*n). - Vaclav Kotesovec, Aug 16 2013
a(n) = (1/Pi^2) * Integral_{y = 0..Pi} Integral_{x = 0..Pi} (2*cos(x) + 2*cos(y) + 4*cos(x)*cos(y))^n dx dy. - Peter Bala, Feb 14 2017
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^2. - Seiichi Manyama, Oct 29 2019
From Peter Bala, Feb 08 2022: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k.
Conjecture: the stronger congruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 5 and positive integers n and k. (End)
a(n) = Sum_{j = 0..n} Sum_{k = 0..j} binomial(2*j,j)^2*binomial(j,k)* binomial(n+j-k,2*j)*(-4)^(n-j-k). - Peter Bala, Mar 19 2022

More terms from and entry improved by Sergey Perepechko, Sep 06 2004

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

A049037 Number of cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.

Original entry on oeis.org

1, 6, 54, 996, 22734, 577692, 15680628, 445162392, 13055851998, 392475442092, 12029082873372, 374482032292008, 11808861461931492, 376406128925067528, 12108063535794336312, 392560994063887113744, 12814685828476778001726, 420836267423433182275404

Offset: 0

Alessandro Zinani (alzinani(AT)tin.it)

			a(5) = 577692 because there are 577692 different walks that start and end at the origin after 2*5=10 steps, avoiding origin at intermediate steps.

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

Alois P. Heinz, Table of n, a(n) for n = 0..200
Steven R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice
Steven R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice [Cached copy, with permission of the author]
N. J. A. Sloane, Transforms

Invert A002896, A094059.

Column k=3 of A361397.

read transforms; t1 := [ seq(A002896(i),i=1..25) ]; INVERTi(t1);
# second Maple program:
b:= proc(n) option remember; `if`(n<2, 5*n+1,
      (2*(2*n-1)*(10*n^2-10*n+3) *b(n-1)
       -36*(n-1)*(2*n-1)*(2*n-3) *b(n-2)) /n^3)
    end:
g:= proc(n) g(n):= `if` (n<1, -1, -add(g(n-i) *b(i), i=1..n)) end:
a:= n-> abs(g(n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 02 2012

(* A002896 : *) b[n_] := b[n] = Binomial[2*n, n]*HypergeometricPFQ[{1/2, -n, -n}, {1, 1}, 4]; max = 32; a[0] = 1; se = Series[ Sum[ a[n] x^(2 n), {n, 1, max}] - 1 + 1/Sum[ b[n]*x^(2 n), {n, 0, max}], {x, 0, max}]; coes = CoefficientList[se, x]; sol = First[ Solve[ Thread[ coes == 0]]]; Table[ a[n], {n, 0, 16}] /. sol (* Jean-François Alcover, Dec 20 2011 *)
b[n_] := b[n] = If[n < 2, 5*n + 1, (2*(2*n - 1)*(10*n^2 - 10*n + 3)*b[n-1] - 36*(n - 1)*(2*n - 1)*(2*n - 3)*b[n-2]) / n^3];
g[n_] := g[n] = If[n < 1, -1, -Sum [g[n - i]*b[i], {i, 1, n}]];
a[n_] := Abs[g[n]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 12 2018, after Alois P. Heinz *)

Define a_0, a_1, ... = [ 1, 6, 54, ... ] by 1+Sum b_i x^i = 1/(1-Sum a_i x^i) where b_0, b_1, ... = [ 1, 6, 90, ... ] = A002896.
Or, Sum[ a(n) x^(2n), n=1, 2, ...infinity ] = 1-1/Sum[ A002896(n)*x^(2n), n=0, 1, ...infinity ].
G.f.: 2-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 30 2011
a(n) ~ c * 36^n / n^(3/2), where c = 0.1014559485279103938501072426734... . - Vaclav Kotesovec, Sep 13 2014
c = 384 * (3 + 2*sqrt(3)) * Pi^(9/2) / (Gamma(1/24)^4 * Gamma(11/24)^4). - Vaclav Kotesovec, Apr 23 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

A063887 Number of n-step walks on a square lattice starting from the origin but not returning to it at any stage.

Original entry on oeis.org

1, 4, 12, 48, 172, 688, 2576, 10304, 39340, 157360, 607376, 2429504, 9442448, 37769792, 147495104, 589980416, 2311926188, 9247704752, 36333781776, 145335127104, 572189853200, 2288759412800, 9025822792896, 36103291171584, 142567754881168, 570271019524672, 2254477964009664, 9017911856038656

Offset: 0

Henry Bottomley, Aug 28 2001

nonn

a(n)/4^n tends to zero as n increases.

			a(2)=12 since there are 16 2-step walks but 4 of them involve a return to the origin at some stage; similarly a(3)=48 since there are 64 3-step walks but 16 of them involve a return to the origin at some stage.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

CoefficientList[ Pi/(2 (1 - 4 x) EllipticK[16 x^2]) + O[x]^25, x] (* Jean-François Alcover, Jun 02 2019 *)

my(x='x+O('x^33)); Vec( agm(1, (1+4*x)/(1-4*x)) ) \\ Joerg Arndt, May 17 2019

a(2n) = 4*a(2n-1) - A054474(n); a(2n+1) = 4*a(2n).
G.f.: agm(1, (1+4*x)/(1-4*x)), where agm(x,y) = agm((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean. - Paul D. Hanna, Oct 03 2014
a(n) ~ Pi * 4^n / log(n) * (1 - (gamma + 3*log(2)) / log(n) + (gamma^2 + 6*gamma*log(2) + 9*log(2)^2 - Pi^2/6) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

a(23) corrected by Jean-François Alcover, Jun 02 2019

A359801 Number of 4-dimensional cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.

Original entry on oeis.org

1, 8, 104, 2944, 108136, 4525888, 204981888, 9792786432, 486323201640, 24874892400064, 1302278744460352, 69474942954714112, 3764568243058030208, 206675027529594291200, 11473858525271117889536, 643154944963894079717376, 36355546411928157876528744, 2070313613815122857027563200

Offset: 0

Shel Kaphan, Mar 08 2023

In Novak's note it is mentioned that if P(z) and Q(z) are the g.f.s for the probabilities of indecomposable and decomposable loops, respectively, that P(z) = 1 - 1/Q(z). This works equally well using loop counts rather than probabilities. The g.f.s may be expressed by the series constructed from the sequences of counts of loops of length 2*n. Q(z) for the 4-d case is the series corresponding to A039699.

To obtain the probability of returning to the point of origin for the first time after 2*n steps, divide a(n) by the total number of walks of length 2*n in d dimensions: (2*d)^(2*n) = 64^n.

Vaclav Kotesovec, Table of n, a(n) for n = 0..550
Dorin Dumitraşcu and Liviu Suciu, Asymptotics for the Number of Random Walks in the Euclidean Lattice, arXiv:2212.01702 [math.CO], 2022, p.11.
Jonathan Novak, Pólya's Random Walk Theorem, The American Mathematical Monthly, Vol. 121, No. 8 (October 2014), pp. 711-716.

Cf. A039699, A287317 (number of walks that return to the origin in 2n steps).
Number of walks that return to the origin for the first time in 2n steps, in 1..3 dimensions: |A002420|, A054474, A049037.
Column k=4 of A361397.

walk4d[n_] :=
 Sum[(2 n)!/(i! j! k! (n - i - j - k)!)^2, {i, 0, n}, {j, 0,
   n - i}, {k, 0, n - i - j}]; invertSeq[seq_] :=
  CoefficientList[1 - 1/SeriesData[x, 0, seq, 0, Length[seq], 1], x]; invertSeq[Table[walk4d[n], {n, 0, 17}]]

seq(n) = {my(v=Vec(2 - 1/serlaplace(besseli(0, 2*x + O(x^(2*n+1)))^4))); vector(n+1, i, v[2*i-1])} \\ Andrew Howroyd, Mar 08 2023

G.f.: 2 - 1/Q(x) where Q(x) is the g.f. of A039699.
G.f.: 2 - 1/Integral_{t=0..oo} exp(-t)*BesselI(0,2*t*sqrt(x))^4 dt.
INVERTi transform of A039699.

A043546 Coefficients of asymptotic expansion of return probability for random walk in d-dimensional cubic lattice as a function of d.

Original entry on oeis.org

0, 1, 2, 7, 35, 215, 1501, 11354, 88978, 675569, 4175664, 1725333, -687775083, -19848956619, -438027976068, -8715988203509, -161989586455204, -2784493824166078, -41530410660307610, -406672888265416456, 4420077014249902362, 456572861717941696791

Offset: 0

Joe Keane (jgk(AT)jgk.org)

sign

			Flajolet's code gives the following asymptotic expansion: p(d) = 1/(2*d) + 2/(2*d)^2 + 7/(2*d)^3 + 35/(2*d)^4 + 215/(2*d)^5 + 1501/(2*d)^6 + 11354/(2*d)^7 + 88978/(2*d)^8 + 675569/(2*d)^9 + 4175664/(2*d)^10 + 1725333/(2*d)^11 - 687775083/(2*d)^12 - 19848956619/(2*d)^13 - 438027976068/(2*d)^14 - 8715988203509/(2*d)^15 - 161989586455204/(2*d)^16 - 2784493824166078/(2*d)^17 - 41530410660307610/(2*d)^18 - 406672888265416456/(2*d)^19 + 4420077014249902362/(2*d)^20 + ...
G.f. = x + 2*x^2 + 7*x^3 + 35*x^4 + 215*x^5 + 1501*x^6 + 11354*x^7 + ...

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

Vincenzo Librandi, Table of n, a(n) for n = 0..60
Noam D. Elkies, Reply to: Pólya's Random Walk Constants at infinity, MathOverflow Q-83317, Dec 13 2011.
S. R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice
Steven R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice [Cached copy, with permission of the author]
P. Flajolet, Reply to: Symmetric random walk on n-dimensional integer lattice, sci.math.research newsgroup posting, 1995.

walk:=proc(order) local n,j:
j:=sum(t^(2*n)/(2*d)^(2*n)/n!^2,n=0..order): eval(subs(O=0, asympt(exp(d*log(j)),d,order+2)))*exp(-t): 1-1/int(%,t=0..infinity):
RETURN(asympt(asympt(%,d,2*order+5),d,order+1)):
end:
seq(coeff(convert(walk(20),polynom),d,-n)*2^n,n=0..20); (Ronaldo)

nn = 20; I1 = Sum[x^n/n!^2, {n, 0, nn}]; Iw = (I1 /. x -> w^2*x)^(1/(2*w)); g = Sum[(2*n)!*SeriesCoefficient[Iw, {x, 0, n}], {n, 0, nn}]; p = 1 - 1/g; Table[SeriesCoefficient[p, {w, 0, n}], {n, 0, nn}] (* Jean-François Alcover, Jan 08 2014, after the PARI code by Noam D. Elkies *)
a[ n_] := If[n < 0, 0, With[{A = Series[ BesselI[ 0, 2 Sqrt[y y x]]^(1/(2 y)), {x, 0, n}]}, SeriesCoefficient[ 1 - 1 / (Sum[(2 k)! SeriesCoefficient[ A, {x, 0, k}], {k, 0, n}]), {y, 0, n}]]]; (* Michael Somos, May 25 2014 *)

N = 20
I1 = sum(n=0,N,x^n/n!^2,O(x^(N+1)));
Iw = subst(I1,x,w^2*x)^(1/(2*w));
g = sum(n=0,N,(2*n)!*polcoeff(Iw,n,x)) + O(w^(N+1));
p = 1 - 1/g
vector(N,n,polcoeff(p,n))
\\ Noam D. Elkies, Dec 13 2011 (see link)

Edited by C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 27 2004

cp's OEIS Frontend

A094059 Analog of A054474 for walks on a 3-dimensional grid.

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A002894 a(n) = binomial(2n, n)^2.

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A094061 Number of n-moves paths of a king starting and ending at the origin of an infinite chessboard.

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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.

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A049037 Number of cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.

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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.

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