A094061 -id:A094061 - cp's OEIS frontend

A054474 Number of walks on square lattice that start and end at origin after 2n steps, not touching origin at intermediate stages.

1, 4, 20, 176, 1876, 22064, 275568, 3584064, 47995476, 657037232, 9150655216, 129214858304, 1845409805168, 26606114089024, 386679996988736, 5658611409163008, 83302885723872852, 1232764004638179504, 18327520881735288432, 273595871825723062848

Offset: 0

Alessandro Zinani (alzinani(AT)tin.it), May 19 2000

1-dimensional and 3-dimensional analogs are A002420 and A049037.
Trajectories returning to the origin are prohibited, contrary to the situation in A094061.
The probability of returning to the origin for the first time after 2n steps is given by a(n)/4^(2*n). If A(x) is a generating function for this sequence, A(x/16) is a generating function for the sequence of probabilities. The sum of these probabilities for n > 0 is 1 unlike in dimensions > 2. - Shel Kaphan, Feb 13 2023

			a(5)=22064, i.e., there are 22064 different walks (on a square lattice) that start and end at the origin after 2*5=10 steps, avoiding the 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..834
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]
Markus Kuba and Alois Panholzer, Lattice paths and the diagonal of the cube, arXiv:2411.03930 [math.CO], 2024. See p. 14.

Cf. A002894, A002420, A049037, A063887.

Column k=2 of A361397.

b:= proc(n) b(n):= binomial(2*n, n)^2 end:
a:= proc(n) option remember;
      b(n)-add(a(n-i)*b(i), i=1..n-1)
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Dec 05 2023

m = 18; gf[x_] = 2 - Pi/(2*EllipticK[4*Sqrt[x]]); (List @@ Normal[ Series[ gf[x], {x, 0, m-1}]] /. x -> 1)[[1 ;; m+1]]*Table[4^k, {k, 0, m}] (* Jean-François Alcover, Jun 16 2011, after Vladeta Jovovic *)
CoefficientList[Series[2-Pi/(2*EllipticK[16*x]),{x,0,20}],x] (* Vaclav Kotesovec, Mar 10 2014 *)
CoefficientList[Series[2-ArithmeticGeometricMean[1,Sqrt[1-16x]],{x,0,20}],x] (* Thomas Dybdahl Ahle, Oct 30 2023 *)

a(n)=if(n<0,0,polcoeff(2-agm(1,sqrt(1-16*x+x*O(x^n))),n))

G.f.: 2 - AGM(1, (1-16*x)^(1/2)).
G.f.: 2 - 1/hypergeom([1/2,1/2],[1],16*x). - Joerg Arndt, Jun 16 2011
Let (in Maple notation) G(x):=4/Pi*EllipticK(4*t)-2/Pi*EllipticF(1/sqrt(2+4*t), 4*t)-2/Pi*EllipticF(1/sqrt(2-4*t), 4*t), then GenFunc(x):=2-1/G(x). - Sergey Perepechko, Sep 11 2004
G.f.: 2 - Pi/(2*EllipticK(4*sqrt(x))). - Vladeta Jovovic, Jun 23 2005
a(n) ~ Pi * 16^n / (n * log(n)^2) * (1 - (2*gamma + 8*log(2)) / log(n) + (3*gamma^2 + 24*log(2)*gamma + 48*log(2)^2 - Pi^2/2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019
INVERTi transform of A002894. - R. J. Mathar, Sep 24 2020

A254129 Number of 2n-move closed knight paths on an unbounded chessboard from a given square to the same square.

Original entry on oeis.org

1, 8, 168, 5840, 261800, 13180608, 702273264, 38641656768, 2171652448680, 123938999632448, 7158206751686848, 417418594698260064, 24535017440445455216, 1451786144317963971200, 86396682439552099487040, 5166936574734171792925440, 310340697572034456203934120

Offset: 0

David A. Corneth, Jan 25 2015

Every move changes the color of the square the knight is on, so there is no returning path of odd length.

			a(1) = 8. For illustration, let's assume we're on a usual 8 X 8 chessboard, with the knight initially at D4. Then there are 8 paths bringing it back to D4 in 2 moves:
D4-E6-D4; D4-F5-D4; D4-F3-D4; D4-E2-D4; D4-C2-D4; D4-B3-D4; D4-B5-D4; D4-C6-D4.

Alois P. Heinz, Table of n, a(n) for n = 0..550 (first 85 terms from David A. Corneth)
Shalosh B. Ekhad and Doron Zeilberger, In How Many Ways Can the Chess Pieces Walk n Steps, Staying on the Board?, May 19 2011; Local copy, pdf file only, no active links
Mohamud Mohammed and Doron Zeilberger, Maple program SMAZ; Local copy
Doron Zeilberger, Input file inSMAZ6; Local copy
Doron Zeilberger, Output from Maple program SMAZ; Local copy

Cf. A025600, A094061, A253974, A254459, A307347.

G:= cos(x+2*y)+cos(x-2*y)+cos(2*x+y)+cos(2*x-y):
F:= 1: a[0]:= 1:
for n from 1 to 20 do
  F:= combine(F*G^2,trig);
  a[n]:= 4^n*remove(has,F,cos);
od:
seq(a[n],n=0..20);  # Robert Israel, Jan 26 2015
# second Maple program:
b:= proc(n, x, y) option remember; `if`(max(x, y)>2*n or x+y>3*n, 0,
      `if`(n=0, 1, add(b(n-1, abs(x+l[1]), abs(y+l[2])), l=[[1, 2],
      [2, 1], [-1, 2], [-2, 1], [1, -2], [2, -1], [-1, -2], [-2, -1]])))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 29 2015

b[n_, x_, y_] := b[n, x, y] = If[Max[x, y] > 2n || x+y > 3n, 0, If[n == 0, 1, Sum[b[n-1, Abs[x+l[[1]]], Abs[y+l[[2]]]], {l, {{1, 2}, {2, 1}, {-1, 2}, {-2, 1}, {1, -2}, {2, -1}, {-1, -2}, {-2, -1}}}]]];
a[n_] :=  b[2n, 0, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 30 2019, after Alois P. Heinz *)

a(n)={my(l=listcreate(),v=vector(2*n+1));m=matrix(1,1);m[1,1]=1;listput(l,m);v[1]=1;for(i=2,2*n+1, m=matrix(4*i-3,4*i-3);for(j=1,#l[i-1],for(k=1,#l[i-1],m[j+2-2,k+2-1]+=l[i-1][j,k];m[j+2-2,k+2+1]+=l[i-1][j,k];m[j+2-1,k+2-2]+=l[i-1][j,k];m[j+2-1,k+2+2]+=l[i-1][j,k];m[j+2+1,k+2-2]+=l[i-1][j,k];m[j+2+1,k+2+2]+=l[i-1][j,k];m[j+2+2,k+2-1]+=l[i-1][j,k];m[j+2+2,k+2+1]+=l[i-1][j,k]));v[i]=m[2*i-1,2*i-1];listput(l,m););listput(l,v);v[#v]} \\ David A. Corneth, Jan 26 2015

{a(n) = polcoef(polcoef((x^2*y+x*y^2+y^2/x+y/x^2+1/(x^2*y)+1/(x*y^2)+x/y^2+x^2/y)^(2*n), 0), 0)} \\ Seiichi Manyama, Nov 02 2019

From Robert Israel, Jan 26 2015: (Start)
a(n) = 4^n * (2*Pi)^(-2) * int_0^(2*Pi) int_0^(2*Pi) ds dt (cos(s+2*t)+cos(s-2*t)+cos(2*s+t)+cos(2*s-t))^(2*n).
G.f.: (2*Pi)^(-2) * int_0^(2*Pi) int_0^(2*Pi) ds dt 1/(1 - 4*x*(cos(s+2*t)+cos(s-2*t)+cos(2*s+t)+cos(2*s-t))^2).
(End)
a(n) ~ 64^n / (5*Pi*n). - Vaclav Kotesovec, Jan 28 2015
Recurrence (conjectured): 3*n^2*(3*n-2)*(3*n-1)*(4*n-3)*(4*n-1)*(58625*n^6 - 574525*n^5 + 2317575*n^4 - 4929815*n^3 + 5836090*n^2 - 3647730*n + 940788)*a(n) = 4*(563444875*n^12 - 7212094400*n^11 + 40894216825*n^10 - 135653664390*n^9 + 292742658975*n^8 - 432166599360*n^7 + 446527351283*n^6 - 324481592710*n^5 + 164046706898*n^4 - 56035458036*n^3 + 12203976528*n^2 - 1507156200*n + 78246000)*a(n-1) - 64*(n-1)*(2*n-3)^2*(167726125*n^9 - 1643716025*n^8 + 6735239425*n^7 - 15048594215*n^6 + 20072439970*n^5 - 16473493280*n^4 + 8273936628*n^3 - 2437948332*n^2 + 377982648*n - 22556880)*a(n-2) + 165888*(n-2)*(n-1)*(2*n - 5)^2*(2*n - 3)^2*(58625*n^6 - 222775*n^5 + 324325*n^4 - 232265*n^3 + 86220*n^2 - 15570*n + 1008)*a(n-3). - Vaclav Kotesovec, Jan 30 2015
a(n) = the constant term in the expansion of (x^2*y + x*y^2 + y^2/x + y/x^2 + 1/(x^2*y) + 1/(y^2*x) + x/y^2 + x^2/y)^(2*n). - Peter Bala, Feb 14 2017
The conjectured recurrence of Vaclav Kotesovec is true. Running the input file inSMAZ6 (see Links) on the Maple program SMAZ gives the recurrence followed by the certificate shown in the output file oSMAZ6.  - Doron Zeilberger, Mar 29 2019

A168597 Squares of the central trinomial coefficients (A002426).

Original entry on oeis.org

1, 1, 9, 49, 361, 2601, 19881, 154449, 1225449, 9853321, 80156209, 658076409, 5444816521, 45343869481, 379735715529, 3195538786449, 27004932177129, 229066136374761, 1949470542590481, 16640188083903609, 142415188146838161, 1221800234100831441, 10504959504381567729

Offset: 0

Paul D. Hanna, Dec 01 2009

Ignoring initial term, a(n) equals the logarithmic derivative of A168598.
Partial sums of A007987. Hence, a(n) is the number of irreducible words of length at most 2n in the free group with generators x,y such that the total degree of x and the total degree of y both equal zero. - Max Alekseyev, Jun 05 2011
The number of ways a king, starting at the origin of an infinite chessboard, can return to the origin in n moves, where leaving the king where it is counts as a move. Cf. A094061. - Peter Bala, Feb 14 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A002426, A133053, A168598, A243949, A094061.

a := n -> (-1)^n*hypergeom([1/2,-n],[1],4)*hypergeom([1/2-n/2,-n/2],[1], 4): seq(simplify(a(n)),n=0..20); # Peter Luschny, Nov 10 2014

Table[(-1)^n*Hypergeometric2F1[1/2, -n, 1, 4] * Hypergeometric2F1[(1 - n)/2, -n/2, 1, 4], {n, 0, 50}] (* G. C. Greubel, Feb 26 2017 *)
CoefficientList[Series[(2 EllipticK[(16 x)/(1 + 3 x)^2])/(Pi (1 + 3 x)), {x, 0, 28}], x, 26]  (* After Mark van Hoeij, Peter Luschny, May 13 2025 *)

{a(n)=polcoeff((1+x+x^2 +x*O(x^n))^n,n)^2}
for(n=0, 20, print1(a(n), ", "))

/* Using AGM: */
{a(n)=polcoeff( 1 / agm(1+3*x, sqrt((1+3*x)^2 - 16*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 04 2014

a(n) = A002426(n)^2.
G.f.: hypergeom([1/12, 5/12],[1],1728*x^4*(x-1)*(9*x-1)*(3*x+1)^2/(81*x^4-36*x^3-26*x^2-4*x+1)^3)/(81*x^4-36*x^3-26*x^2-4*x+1)^(1/4). - Mark van Hoeij, May 07 2013
G.f.: 1 / AGM(1+3*x, sqrt((1-x)*(1-9*x))), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean. - Paul D. Hanna, Sep 04 2014
G.f.: 1 / AGM((1-x)*(1-3*x), (1+x)*(1+3*x)) = Sum_{n>=0} a(n)*x^(2*n). - Paul D. Hanna, Oct 04 2014
a(n) = (-1)^n*hypergeom([1/2,-n],[1],4)*hypergeom([(1-n)/2,-n/2],[1],4). - Peter Luschny, Nov 10 2014
a(n) ~ 3^(2*n+1) / (4*Pi*n). - Vaclav Kotesovec, Sep 28 2019
From Peter Bala, Feb 08 2022: (Start)
a(n) = Sum_{k = 0..n} (-3)^(n-k)*binomial(2*k,k)*binomial(n,k)* binomial(n+k,k).
n^2*(2*n-3)*a(n)= (7*n^2-14*n+6)*(2*n-1)*a(n-1) + 3*(7*n^2-14*n+6)*(2*n-3)*a(n-2) - 27*(2*n-1)*(n-2)^2*a(n-3) with a(0) = 1, a(1) = 1 and a(2) = 9.
G.f.: A(x) = Sum_{n >= 0} binomial(2*n,n)^2*x^n/(1 + 3*x)^(2*n+1).
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all prime 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)
G.f.: hypergeom([1/2, 1/2],[1],16*x/(1+3*x)^2)/(1+3*x). - Mark van Hoeij, May 13 2025

A253974 Number of 2n-move closed giraffe paths on an unbounded chessboard from a given square to the same square.

Original entry on oeis.org

1, 8, 168, 5120, 190120, 7964208, 362370624, 17532536736, 889716433320, 46887220540160, 2546408317827088, 141659449976239104, 8033749056463329472, 462687411167492828000, 26980019699392099317600, 1589091557661690119997120, 94361786346423775855372200

Offset: 0

Vaclav Kotesovec, Jan 31 2015

nonn

Giraffe is a (fairy chess) leaper [1,4].

Conjecture: Number of 2n-move closed paths of leaper [r,s] on an unbounded chessboard, where 0 < r < s and gcd(r,s)=1, is asymptotic to 2^(6*n+1) / ((r^2+s^2)*Pi*n) if r+s is even, and 2^(6*n) / ((r^2+s^2)*Pi*n) if r+s is odd.

Vaclav Kotesovec, Table of n, a(n) for n = 0..552
Vaclav Kotesovec, Conjectured recurrence (of order 8)
Vaclav Kotesovec, Examples of closed giraffe paths
Wikipedia, Fairy chess piece

Cf. A094061, A254129, A254459, A307347.

b:= proc(n, x, y) option remember; `if`(max(x, y)>4*n or x+y>5*n, 0,
      `if`(n=0, 1, add(b(n-1, abs(x+l[1]), abs(y+l[2])), l=[[4, 1],
      [1, 4], [-4, 1], [-1, 4], [4, -1], [1, -4], [-4, -1], [-1, -4]])))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..25); # after Alois P. Heinz
# second Maple program:
poly:=expand((x*y^4+x^4*y+y^4/x+y/x^4+x/y^4+x^4/y+1/(x*y^4)+1/(x^4*y))^2): z:=1: for n to 100 do z:=expand(z*poly): print(n, coeff(coeff(z, x, 0), y, 0)); end do: # Vaclav Kotesovec, Apr 03 2019

b[n_, x_, y_] := b[n, x, y] = If[Max[x, y] > 4n || x + y > 5n, 0, If[n == 0, 1, Sum[b[n - 1, Abs[x + l[[1]]], Abs[y + l[[2]]]], {l, {{4, 1}, {1, 4}, {-4, 1}, {-1, 4}, {4, -1}, {1, -4}, {-4, -1}, {-1, -4}}}]]];
a[n_] := b[2n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 01 2020, after Maple *)

a(n) ~ 64^n / (17*Pi*n).

a(n) = the constant term in the expansion of (x*y^4 + x^4*y + 1/x*y^4 + 1/x^4*y + x/y^4 + x^4/y + 1/x/y^4 + 1/x^4/y)^(2*n). - Vaclav Kotesovec, Apr 01 2019

A254459 Number of 2n-move closed zebra paths on an unbounded chessboard from a given square to the same square.

Original entry on oeis.org

1, 8, 168, 5120, 190120, 8039808, 373369920, 18576523680, 972362837160, 52832252432960, 2950644716576128, 168192125309339040, 9735527029198105408, 570163460613978204800, 33697054064651581144800, 2005939326990647575285920, 120109818840839172931095720

Offset: 0

Vaclav Kotesovec, Jan 30 2015

nonn

Zebra is a (fairy chess) leaper [2,3].

Conjecture: Number of 2n-move closed paths of leaper [r,s] on an unbounded chessboard, where 0 < r < s and gcd(r,s)=1, is asymptotic to 2^(6*n+1) / ((r^2+s^2)*Pi*n) if r+s is even, and 2^(6*n) / ((r^2+s^2)*Pi*n) if r+s is odd.

Vaclav Kotesovec, Table of n, a(n) for n = 0..542
Vaclav Kotesovec, Examples of closed zebra paths
Vaclav Kotesovec, Conjectured recurrence (of order 6)
Wikipedia, Fairy chess piece

Cf. A094061, A253974, A254129, A307347.

b:= proc(n, x, y) option remember; `if`(max(x, y)>3*n or x+y>5*n, 0,
      `if`(n=0, 1, add(b(n-1, abs(x+l[1]), abs(y+l[2])), l=[[3, 2],
      [2, 3], [-3, 2], [-2, 3], [3, -2], [2, -3], [-3, -2], [-2, -3]])))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..25); # after Alois P. Heinz
# second Maple program:
poly:=expand((x^2*y^3 + x^3*y^2 + 1/x^2*y^3 + 1/x^3*y^2 + x^2/y^3 + x^3/y^2 + 1/x^2/y^3 + 1/x^3/y^2)^2): z:=1: for n to 100 do z:=expand(z*poly): print(n, coeff(coeff(z, x, 0), y, 0)); end do: # Vaclav Kotesovec, Apr 03 2019

b[n_, x_, y_] := b[n, x, y] = If[Max[x, y] > 3n || x + y > 5n, 0, If[n == 0, 1, Sum[b[n - 1, Abs[x + l[[1]]], Abs[y + l[[2]]]], {l, {{3, 2}, {2, 3}, {-3, 2}, {-2, 3}, {3, -2}, {2, -3}, {-3, -2}, {-2, -3}}}]]];
a[n_] := b[2n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 01 2020, after Maple *)

a(n) ~ 64^n / (13*Pi*n).

a(n) = the constant term in the expansion of (x^2*y^3 + x^3*y^2 + 1/x^2*y^3 + 1/x^3*y^2 + x^2/y^3 + x^3/y^2 + 1/x^2/y^3 + 1/x^3/y^2)^(2*n). - Vaclav Kotesovec, Apr 01 2019

A288470 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2n,2*k).

Original entry on oeis.org

1, 2, 14, 92, 646, 4652, 34124, 253528, 1901638, 14368844, 109208164, 833981128, 6394017436, 49185717752, 379438594136, 2934361958192, 22741538394694, 176582855512588, 1373431963785332, 10698376362421096, 83447762846703796, 651690159076273192, 5095051571420324264, 39874449115469939152, 312350761370734541596

Offset: 0

Robert Israel, Jun 09 2017

Row sums of A155495.

a(n) is the constant term in the expansion of (-1 + (1 + x + 1/x)^2)^n. - Seiichi Manyama, Nov 21 2019

Seiichi Manyama, Table of n, a(n) for n = 0..1109

Cf. A094061, A155495, A218045.

f:= gfun:-rectoproc({n*(2*n-1)*a(n) = (32*(n-2))*(2*n-5)*a(n-3)+(8*(9*n^2-31*n+28))*a(n-2)+(2*(3*n^2+7*n-9))*a(n-1), a(0)=1,a(1)=2, a(2)=14},a(n),remember):
map(f, [$0..30]);

Table[Sum[Binomial[n, k] Binomial[2 n, 2 k], {k, 0, n}], {n, 0, 24}] (* Michael De Vlieger, Jun 09 2017 *)

{a(n) = polcoef((-1+(1+x+1/x)^2)^n, 0)} \\ Seiichi Manyama, Nov 21 2019

a(n) = hypergeom([-n,-n,1/2-n],[1/2,1],-1).
n*(2*n-1)*a(n) = (32*(n-2))*(2*n-5)*a(n-3)+(8*(9*n^2-31*n+28))*a(n-2)+(2*(3*n^2+7*n-9))*a(n-1).
G.f.: sqrt((1-2*x+sqrt(1-8*x))/(2*(1-7*x-8*x^2))).
a(n) ~ 8^n / sqrt(3*Pi*n). - Vaclav Kotesovec, Nov 27 2017
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k) * binomial(3*n-2*k-1,n-2*k). - Seiichi Manyama, Feb 13 2024
From Peter Bala, Aug 30 2025: (Start)
n*(2*n - 1)*(3*n - 4)*a(n) = 2*(21*n^3 - 49*n^2 + 33*n - 6)*a(n-1) + 8*(n - 1)*(3*n - 1)*(2*n - 3)*a(n-2) with a(0) = 1 and a(1) = 2.
a(n) = Sum_{0 <= i, j <= n/2} binomial(2*n, j)*binomial(2*n+i-1, i)*binomial(2*n, n- 2*i-2*j) (verified to satisfy the above second-order recurrence using the MulZeil procedure in Doron Zeilberger's MultiZeilberger Maple package).
Equivalently, a(n) = [x^n] ( (1 + x + x^2 + x^3)/(1 - x^2) )^(2*n). Cf. A240688.
The Gauss congruences hold: a(n*p^k) == a(n*p^(k-1)) (mod p^k)  for all primes p and positive integers n and k.
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for primes p >= 5 and positive integers n and k. (End)

A328874 Constant term in the expansion of (-1 + (1 + x + 1/x) * (1 + y + 1/y) * (1 + z + 1/z))^n.

Original entry on oeis.org

1, 0, 26, 264, 5646, 101520, 2103740, 43632960, 942507790, 20685977760, 462661368876, 10483696885200, 240373512418116, 5564581640601984, 129901678525143096, 3054381796821779424, 72272856926974596750, 1719662128611006026304, 41120565854695068532076, 987633314722818034066224

Offset: 0

Seiichi Manyama, Oct 29 2019

nonn

Also number of n-step closed walks (from origin to origin) in cubic lattice, using steps (t_1,t_2,t_3) (t_k = -1, 1 or 0 for 1 <= k <= 3) except for (0,0,0).

Seiichi Manyama, Table of n, a(n) for n = 0..710

Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^m: A126869 (m=1), A094061 (m=2), this sequence (m=3), A328875 (m=4).

Cf. A326920.

Table[Sum[(-1)^(n-k) * Binomial[n, k] * Sum[Binomial[k, 2*j]*Binomial[2*j, j], {j, 0, k}]^3, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 30 2019 *)
Table[Sum[(-1)^(n-k) * Binomial[n, k] * Hypergeometric2F1[1/2 - k/2, -k/2, 1, 4]^3, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 30 2019 *)

{a(n) = polcoef(polcoef(polcoef((-1+(1+x+1/x)*(1+y+1/y)*(1+z+1/z))^n, 0), 0), 0)}

{a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef((1+x+1/x)^k, 0)^3)}

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^3.
From Vaclav Kotesovec, Oct 30 2019: (Start)
Recurrence: (n-1)*n^3*(5103*n^4 - 38556*n^3 + 107838*n^2 - 132564*n + 60401)*a(n) = (n-1)^2*(71442*n^6 - 575505*n^5 + 1817613*n^4 - 2850549*n^3 + 2299999*n^2 - 891084*n + 132528)*a(n-1) + (1505385*n^8 - 17395560*n^7 + 85857516*n^6 - 235935678*n^5 + 393710399*n^4 - 407039414*n^3 + 253464484*n^2 - 86477832*n + 12324048)*a(n-2) + 2*(n-2)*(1224720*n^7 - 13539960*n^6 + 61400268*n^5 - 146649411*n^4 + 197630220*n^3 - 149760433*n^2 + 59083626*n - 9168258)*a(n-3) - 4*(n-3)*(n-2)*(1153278*n^6 - 11020212*n^5 + 40809852*n^4 - 74540514*n^3 + 70559711*n^2 - 32643654*n + 5797748)*a(n-4) - 8*(n-4)*(n-3)*(n-2)*(1367604*n^5 - 9649206*n^4 + 23421096*n^3 - 25438791*n^2 + 12638258*n - 2271566)*a(n-5) - 1040*(n-5)*(n-4)*(n-3)*(n-2)*(5103*n^4 - 18144*n^3 + 22788*n^2 - 12144*n + 2222)*a(n-6).
a(n) ~ 13 * 26^(n + 1/2) / (108 * Pi^(3/2) * n^(3/2)). (End)

A329074 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ((Sum_{j=-n..n} x^j) * (Sum_{j=-n..n} y^j) - (Sum_{j=-n+1..n-1} x^j) * (Sum_{j=-n+1..n-1} y^j))^k.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 8, 0, 1, 1, 24, 16, 0, 1, 1, 216, 48, 24, 0, 1, 1, 1200, 1200, 72, 32, 0, 1, 1, 8840, 10200, 3336, 96, 40, 0, 1, 1, 58800, 165760, 34800, 7008, 120, 48, 0, 1, 1, 423640, 2032800, 912840, 82800, 12600, 144, 56, 0, 1

Offset: 0

Seiichi Manyama, Nov 03 2019

T(n,k) is the number of k-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 2*n).

T(n,k) is the constant term in the expansion of (Sum_{j=0..2*n} (x^j + 1/x^j)*(y^(2*n-j) + 1/y^(2*n-j)) - x^(2*n) - 1/x^(2*n) - y^(2*n) - 1/y^(2*n))^k for n > 0.

			Square array begins:
   1, 1,  1,   1,     1,      1, ...
   1, 0,  8,  24,   216,   1200, ...
   1, 0, 16,  48,  1200,  10200, ...
   1, 0, 24,  72,  3336,  34800, ...
   1, 0, 32,  96,  7008,  82800, ...
   1, 0, 40, 120, 12600, 162000, ...

Seiichi Manyama, Antidiagonals n = 0..100, flattened
Wikipedia, Taxicab geometry.

Rows n=0-3 give A000012, A094061, A329075, A329077.
Main diagonal gives A329076.
Cf. A329066.

{T(n, k) = if(n==0, 1, polcoef(polcoef((sum(j=0, 2*n, (x^j+1/x^j)*(y^(2*n-j)+1/y^(2*n-j)))-x^(2*n)-1/x^(2*n)-y^(2*n)-1/y^(2*n))^k, 0), 0))}

f(n) = (x^(n+1)-1/x^n)/(x-1);
T(n, k) = if(n==0, 1, sum(j=0, k, (-1)^(k-j)*binomial(k, j)*polcoef(f(n)^j*f(n-1)^(k-j), 0)^2))

T(0,k) = 1^k = 1.

See the second code written in PARI.

A328875 Constant term in the expansion of (-1 + (1 + w + 1/w) * (1 + x + 1/x) * (1 + y + 1/y) * (1 + z + 1/z))^n.

Original entry on oeis.org

1, 0, 80, 2160, 121200, 6136800, 356570960, 21225304800, 1321586558320, 84398804078400, 5518934916677280, 367489108030524480, 24852668879410144080, 1702677155195779963200, 117960677109321028039200, 8251450286371615261498560, 582087494621171173360817520

Offset: 0

Seiichi Manyama, Oct 29 2019

nonn

Also number of n-step closed walks (from origin to origin) in 4-dimensional lattice, using steps (t_1,t_2,t_3,t_4) (t_k = -1, 1 or 0 for 1 <= k <= 4) except for (0,0,0,0).

For fixed m > 1, Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^m ~ (3^m - 1)^(n + m/2) / (2^m * 3^(m*(m-1)/2) * Pi^(m/2) * n^(m/2)). - Vaclav Kotesovec, Oct 30 2019

Seiichi Manyama, Table of n, a(n) for n = 0..528

Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^m: A126869 (m=1), A094061 (m=2), A328874 (m=3), this sequence (m=4).

Cf. A326920.

Table[Sum[(-1)^(n-k) * Binomial[n, k] * Sum[Binomial[k, 2*j]*Binomial[2*j, j], {j, 0, k}]^4, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 30 2019 *)

{a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef((1+x+1/x)^k, 0)^4)}

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^4.

a(n) ~ 5 * 80^(n+1) / (729 * Pi^2 * n^2). - Vaclav Kotesovec, Oct 30 2019

A329024 Constant term in the expansion of ((x^3 + x + 1/x + 1/x^3)(y^3 + y + 1/y + 1/y^3) - (x + 1/x)(y + 1/y))^(2*n).

Original entry on oeis.org

1, 12, 588, 49440, 5187980, 597027312, 71962945824, 8923789535232, 1128795397492620, 144940851928720848, 18832163401980525168, 2470451402766989534256, 326667449725835512275488, 43485599433527022301377600, 5821983056232777427055717760

Offset: 0

Seiichi Manyama, Nov 02 2019

nonn

Also number of (2*n)-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 3).
         *
         |
      *-- --*
      |  |  |
   *-- -- -- --*
   |  |  |  |  |
*-- -- --P-- -- --*
   |  |  |  |  |
   *-- -- -- --*
      |  |  |
      *-- --*
         |
         *
Point P move to any position of * in the next step.

Seiichi Manyama, Table of n, a(n) for n = 0..400 (terms 0..185 from Vaclav Kotesovec)
Vaclav Kotesovec, Recurrence of order 4 (conjectured)

Row n=1 of A329066.

Cf. A002894, A094061, A254129.

{a(n) = polcoef(polcoef(((x^3+x+1/x+1/x^3)*(y^3+y+1/y+1/y^3)-(x+1/x)*(y+1/y))^(2*n), 0), 0)}

{a(n) = polcoef(polcoef((sum(k=0, 3, (x^k+1/x^k)*(y^(3-k)+1/y^(3-k)))-x^3-1/x^3-y^3-1/y^3)^(2*n), 0), 0)}

f(n) = (x^(2*n+2)-1/x^(2*n+2))/(x-1/x);
a(n) = sum(k=0, 2*n, (-1)^k*binomial(2*n, k)*polcoef(f(1)^k*f(0)^(2*n-k), 0)^2)

Conjecture: a(n) ~ 3 * 144^n / (19*Pi*n). - Vaclav Kotesovec, Nov 04 2019

cp's OEIS Frontend

A054474 Number of walks on square lattice that start and end at origin after 2n steps, not touching origin at intermediate stages.

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A254129 Number of 2n-move closed knight paths on an unbounded chessboard from a given square to the same square.

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A168597 Squares of the central trinomial coefficients (A002426).

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A253974 Number of 2n-move closed giraffe paths on an unbounded chessboard from a given square to the same square.

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A254459 Number of 2n-move closed zebra paths on an unbounded chessboard from a given square to the same square.

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A288470 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*n,2*k).

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A328874 Constant term in the expansion of (-1 + (1 + x + 1/x) * (1 + y + 1/y) * (1 + z + 1/z))^n.

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A288470 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2n,2*k).

A329024 Constant term in the expansion of ((x^3 + x + 1/x + 1/x^3)(y^3 + y + 1/y + 1/y^3) - (x + 1/x)(y + 1/y))^(2*n).