A094061 -id:A094061 - cp's OEIS frontend

A329067 Constant term in the expansion of ((x^5 + x^3 + x + 1/x + 1/x^3 + 1/x^5)(y^5 + y^3 + y + 1/y + 1/y^3 + 1/y^5) - (x^3 + x + 1/x + 1/x^3)(y^3 + y + 1/y + 1/y^3))^(2*n).

1, 20, 2100, 423440, 117234740, 36938855520, 12321942357648, 4240628338620960, 1489773976776270900, 531369088429408040240, 191788135117910898767200, 69889981814391283195249872, 25671987914195551303751107472, 9493180954173722971961114187200

Offset: 0

Seiichi Manyama, Nov 03 2019

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| = 5).

Seiichi Manyama, Table of n, a(n) for n = 0..300 (terms 0..100 from Vaclav Kotesovec)
Wikipedia, Taxicab geometry.

Row n=2 of A329066.

Cf. A094061, A329075.

{a(n) = polcoef(polcoef(((x^5+x^3+x+1/x+1/x^3+1/x^5)*(y^5+y^3+y+1/y+1/y^3+1/y^5)-(x^3+x+1/x+1/x^3)*(y^3+y+1/y+1/y^3))^(2*n), 0), 0)}

{a(n) = polcoef(polcoef((sum(k=0, 5, (x^k+1/x^k)*(y^(5-k)+1/y^(5-k)))-x^5-1/x^5-y^5-1/y^5)^(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(2)^k*f(1)^(2*n-k), 0)^2)

Conjecture: a(n) ~ 400^n / (17*Pi*n). - Vaclav Kotesovec, Nov 04 2019

A307347 Number of 2n-move closed antelope paths on an unbounded chessboard from a given square to the same square.

Original entry on oeis.org

1, 8, 168, 5120, 190120, 7939008, 357713664, 17010543264, 842994009000, 43192225007360, 2275378947981568, 122724475613935104, 6753785574641857024, 378138077830110886400, 21486835143540141873120, 1236506847203439155401920, 71934214120446285067176360

Offset: 0

Vaclav Kotesovec, Apr 03 2019

nonn

Antelope is a leaper [3,4].

Vaclav Kotesovec, Table of n, a(n) for n = 0..552

Cf. A094061, A253974, A254129, A254459.

b:= proc(n, x, y) option remember; `if`(max(x, y)>4*n or x+y>7*n, 0,
      `if`(n=0, 1, add(b(n-1, abs(x+l[1]), abs(y+l[2])), l=[[4, 3],
      [3, 4], [-4, 3], [-3, 4], [4, -3], [3, -4], [-4, -3], [-3, -4]])))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..25);
# second Maple program:
poly := expand((x^4*y^3 + x^3*y^4 + 1/x^4*y^3 + 1/x^3*y^4 + x^4/y^3 + x^3/y^4 + 1/x^4/y^3 + 1/x^3/y^4)^2): z:=1: for n to 100 do z:=expand(z*poly): print(n, coeff(coeff(z, x, 0), y, 0)); end do:

poly = Expand[(x^4*y^3 + x^3*y^4 + 1/x^4*y^3 + 1/x^3*y^4 + x^4/y^3 + x^3/y^4 + 1/x^4/y^3 + 1/x^3/y^4)^2]; z = 1; Flatten[{1, Table[z = Expand[z*poly]; z[[1]], {n, 1, 15}]}]

a(n) = the constant term in the expansion of (x^4*y^3 + x^3*y^4 + 1/x^4*y^3 + 1/x^3*y^4 + x^4/y^3 + x^3/y^4 + 1/x^4/y^3 + 1/x^3/y^4)^(2*n).

Conjecture: a(n) ~ 64^n / (25*Pi*n).

A326920 Constant term in the expansion of (-1 + Product_{k=1..n} (1 + x_k + 1/x_k))^n.

Original entry on oeis.org

1, 0, 8, 264, 121200, 332810400, 7753173594200, 1440193875113407680, 2250630808138439243100640, 29565964235758317208187044137600, 3307988125501026209547184198622507128848, 3165738749695300492286911657015518806826344524560

Offset: 0

Seiichi Manyama, Oct 29 2019

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..47

Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^m: A126869 (m=1), A094061 (m=2), A328874 (m=3), A328875 (m=4).

Table[Sum[(-1)^(n-k) * Binomial[n, k] * Sum[Binomial[k, 2*j]*Binomial[2*j, j], {j, 0, k}]^n, {k, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Oct 30 2019 *)

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

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

a(n) ~ 3^(n^2 + n/2) / (exp(3/16) * 2^n * Pi^(n/2) * n^(n/2)). - Vaclav Kotesovec, Oct 30 2019

A329075 Constant term in the expansion of ((Sum_{k=-2..2} x^k) * (Sum_{k=-2..2} y^k) - (Sum_{k=-1..1} x^k) * (Sum_{k=-1..1} y^k))^n.

Original entry on oeis.org

1, 0, 16, 48, 1200, 10200, 165760, 2032800, 30115120, 417189360, 6116225976, 88579001280, 1308168101856, 19335388664592, 288264711738432, 4311842765438208, 64819095869951280, 977630677389002208, 14796595755047824432, 224583060859608559680, 3417918348978709970680

Offset: 0

Seiichi Manyama, Nov 03 2019

Also number of n-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 4).

Seiichi Manyama, Table of n, a(n) for n = 0..500 (terms 0..300 from Vaclav Kotesovec)
Vaclav Kotesovec, Recurrence of order 8 (conjectured)
Wikipedia, Taxicab geometry.

Row n=2 of A329074.

Cf. A094061, A329067.

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

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

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

Conjecture: a(n) ~ 2 * 16^n / (11*Pi*n). - Vaclav Kotesovec, Nov 04 2019

A329076 Constant term in the expansion of ((Sum_{k=-n..n} x^k) * (Sum_{k=-n..n} y^k) - (Sum_{k=-n+1..n-1} x^k) * (Sum_{k=-n+1..n-1} y^k))^n.

Original entry on oeis.org

1, 0, 16, 72, 7008, 162000, 17555520, 1093527120, 140846184640, 16016249944800, 2550757928818680, 419682645514181280, 82389928294166805312, 17418502084657134228768, 4123280170924828458697152, 1054943518137131171386437600, 293933660095874311773617934720, 87968971083026619734709639853632

Offset: 0

Seiichi Manyama, Nov 04 2019

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

Seiichi Manyama, Table of n, a(n) for n = 0..150 (terms 0..53 from Vaclav Kotesovec)
Wikipedia, Taxicab geometry.

Main diagonal of A329074.

Cf. A094061, A286928.

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

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

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

Conjecture: a(n) ~ 3 * 2^(3*n - 2) * n^(n-3) / Pi. - Vaclav Kotesovec, Nov 05 2019

A329816 Triangular array, read by rows: T(n,k) = [(xy)^k] (-1 + (1 + x + 1/x)(1 + y + 1/y))^n for -n <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 8, 2, 1, 1, 6, 27, 24, 27, 6, 1, 1, 12, 70, 132, 216, 132, 70, 12, 1, 1, 20, 155, 480, 1070, 1200, 1070, 480, 155, 20, 1, 1, 30, 306, 1370, 4035, 6900, 8840, 6900, 4035, 1370, 306, 30, 1, 1, 42, 553, 3332, 12621, 29750, 51065, 58800, 51065, 29750, 12621, 3332, 553, 42, 1

Offset: 0

Seiichi Manyama, Nov 21 2019

Also the coefficient of (x/y)^k in the expansion of (-1 + (1 + x + 1/x)*(1 + y + 1/y))^n for -n <= k <= n.

T(n,k) is the number of n step walks a chess king can take from (0,0) to (k,k). For example, for n=3 starting from (0,0) there is 1 walk to (3,3), 6 walks to (2,2), 27 walks to (1,1), 24 walks to (0,0), 27 walks to (-1,-1), 6 walks to (-2,-2) and 1 walk to (-3,-3). - Martin Clever, May 27 2023

			-1 + (1 + x + 1/x)*(1 + y + 1/y) = x*y + 1/(x*y) + x/y + y/x + x + 1/x + y + 1/y. So T(1,-1) = 1, T(1,0) = 0, T(1,1) = 1.
Triangle begins:
                          1;
                    1,    0,    1;
              1,    2,    8,    2,   1;
         1,   6,   27,   24,   27,   6,   1;
    1,  12,  70,  132,  216,  132,  70,  12,  1;
1, 20, 155, 480, 1070, 1200, 1070, 480, 155, 20, 1;

Seiichi Manyama, Rows n = 0..50, flattened

T(n,0) gives A094061.
Row sums give A288470.
Cf. A260492, A329819, A329820.

{T(n, k) = polcoef(polcoef((-1+(1+x+1/x)*(1+y+1/y))^n, k), k)}

T(n,k) = T(n,-k).

A098070 Consider a single king on an infinite chessboard. This sequence gives number of n-move paths when king starting at origin reaches the origin again for the first time at step n.

Original entry on oeis.org

1, 0, 8, 24, 152, 816, 5320, 33840, 229144, 1560864, 10906576, 76962912, 550406472, 3969725856, 28875757200, 211436151456, 1557623566104, 11533972310976, 85802992349344, 640901090847360, 4804716170926672, 36138383022850368, 272621594933332000

Offset: 0

Sergey Perepechko, Sep 13 2004

nonn

Traditionally for the "first passage time" problems use initial condition Gf(0)=0, but here we define Gf(0)=1 to make this sequence consistent with similar sequences already present in the database.

			From _Jesiah Darnell_, Sep 22 2023: (Start)
A094061(4) - (a(1)a(3)*2 + a(2)*a(2)*1) = 216 - (0 + 64) = 152, so a(4) = 152.
A094061(7) - (a(1)a(6)*2 + a(2)*a(2)*a(3)*3 + a(2)*a(5)*2 + a(4)*a(3)*2) = 58800 - (0 + 4608 + 13056 + 7296) = 33840, so a(7) = 33840. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..350 from Alois P. Heinz)

Cf. A094061, A054474.

G:=t->2-Pi*(1+4*t)/2/EllipticK(4*sqrt(t*(1+t))/(1+4*t)); Gf:=convert(series(G(t),t,30),polynom): seq(print(i,coeff(Gf,t,i)),i=0..degree(Gf));

CoefficientList[Series[2-Pi/2*(1+4*x)/EllipticK[16*x*(1+x)/(1+4*x)^2],{x,0,22}],x] (* Vaclav Kotesovec, Mar 10 2014 *)

G.f.: 2-Pi/2*(1+4*x)/EllipticK(4*sqrt(x*(1+x))/(1+4*x)), (Maple notation).
G.f.: 2 - AGM(sqrt(1 - 8*x), 1 + 4*x). - Vaclav Kotesovec, Sep 30 2019
a(n) ~ 3*Pi*2^(3*n-1) / (n*log(n)^2) * (1 - 2*(gamma + 2*log(2) + 2*log(3)) / log(n) + (3*gamma^2 + 12*log(2)*gamma + 12*gamma*log(3) + 24*log(2)*log(3) + 12*log(2)^2 + 12*log(3)^2 - Pi^2/2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 30 2019
G.f.: 2 - 1/B(x) where B(x) is the g.f. of A094061. - Jesiah Darnell, Sep 22 2023

A327751 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of (-1 + Product_{j=1..n} (1 + x_j + 1/x_j))^k.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 8, 0, 1, 0, 6, 24, 26, 0, 1, 0, 0, 216, 264, 80, 0, 1, 0, 20, 1200, 5646, 2160, 242, 0, 1, 0, 0, 8840, 101520, 121200, 16080, 728, 0, 1, 0, 70, 58800, 2103740, 6136800, 2410326, 115464, 2186, 0, 1

Offset: 0

Seiichi Manyama, Oct 30 2019

T(n,k) is the number of k-step closed walks (from origin to origin) in n-dimensional lattice, using steps (t_1,t_2, ... ,t_n) (t_j = -1, 1 or 0 for 1 <= j <= n) except for (0,0, ... ,0) (t_j = 0 for 1 <= j <= n).

			Square array begins:
   1, 0,   0,     0,       0,         0, ...
   1, 0,   2,     0,       6,         0, ...
   1, 0,   8,    24,     216,      1200, ...
   1, 0,  26,   264,    5646,    101520, ...
   1, 0,  80,  2160,  121200,   6136800, ...
   1, 0, 242, 16080, 2410326, 332810400, ...

Seiichi Manyama, Antidiagonals n = 0..93, flattened

Columns k=0-3 give A000012, A000004, A024023, 24*A016212(n-2).
Rows n=0-4 give A000007, A126869, A094061, A328874, A328875.
Main diagonal is A326920.
Cf. A002426, A328718.

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

A329077 Constant term in the expansion of ((Sum_{k=-3..3} x^k) * (Sum_{k=-3..3} y^k) - (Sum_{k=-2..2} x^k) * (Sum_{k=-2..2} y^k))^n.

Original entry on oeis.org

1, 0, 24, 72, 3336, 34800, 912840, 15661520, 355423880, 7241240160, 160151370624, 3461028611040, 76789098028104, 1700195813892576, 38037857914721808, 853169553940415712, 19240825799184080520, 435267116844063531456, 9882232970998312871232

Offset: 0

Seiichi Manyama, Nov 04 2019

Also number of n-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 6).

Seiichi Manyama, Table of n, a(n) for n = 0..500 (terms 0..230 from Vaclav Kotesovec)
Wikipedia, Taxicab geometry.

Row n=3 of A329074.

Cf. A002894, A094061, A329024, A329067, A329075.

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

{a(n) = polcoef(polcoef((sum(k=0, 6, (x^k+1/x^k)*(y^(6-k)+1/y^(6-k)))-x^6-1/x^6-y^6-1/y^6)^n, 0), 0)}

f(n) = (x^(n+1)-1/x^n)/(x-1);
a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef(f(3)^k*f(2)^(n-k), 0)^2)

Conjecture: a(n) ~ 6 * 24^n / (73*Pi*n). - Vaclav Kotesovec, Nov 04 2019

A218274 Number of n-step paths from (0,0) to (1,0) where all diagonal, vertical and horizontal steps are allowed.

Original entry on oeis.org

0, 1, 4, 27, 168, 1140, 7800, 54845, 390320, 2815344, 20494320, 150442908, 1111782672, 8264558016, 61743361680, 463306724595, 3489942222624, 26378657835816, 199991245341888, 1520403553182800, 11587257160313120, 88506896001503616, 677426230547667744

Offset: 0

Jon Perry, Nov 01 2012

Equivalent to which linear combinations of (-1,-1), (-1,0), (-1,1), (0,1), (0,-1), (1,1), (1,0), (1,-1) equal (1,0).

			a(2) = 4 because we have [0,1]+[1,-1], [1,1]+[0,-1] and the y-negatives [0,-1]+[1,1], [1,-1]+[0,1].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A094061.

a:= proc(n) option remember; `if`(n<3, n^2,
      ((9*n^4-9*n^3-8*n^2+4*n) *a(n-1)
      +4*(n-1)*(27*n^3-84*n^2+80*n-21) *a(n-2)
      +32*(3*n-1)*(n-1)*(n-2)^2 *a(n-3))/ (n*(n-1)*(n+1)*(3*n-4)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 02 2012

a[n_] := a[n] = If[n<3, n^2,
     ((9n^4-9n^3-8n^2+4n) a[n-1] +
     4(n-1)(27n^3-84n^2+80n-21) a[n-2] +
     32(3n-1)(n-1)(n-2)^2 a[n-3]) /
     (n(n-1)(n+1)(3n-4))];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 29 2021, after Alois P. Heinz *)

a[0]:0$
a[1]:1$
a[2]:4$
a[n]:= ((9*n^4-9*n^3-8*n^2+4*n)*a[n-1]+4*(n-1)*(27*n^3-84*n^2+80*n-21)*a[n-2]+32*(3*n-1)*(n-1)*(n-2)^2 *a[n-3])/(n*(n-1)*(n+1)*(3*n-4))$
A218274(n):=a[n]$
makelist(A218274(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

More terms from Joerg Arndt, Nov 02 2012

cp's OEIS Frontend

A329067 Constant term in the expansion of ((x^5 + x^3 + x + 1/x + 1/x^3 + 1/x^5)*(y^5 + y^3 + y + 1/y + 1/y^3 + 1/y^5) - (x^3 + x + 1/x + 1/x^3)*(y^3 + y + 1/y + 1/y^3))^(2*n).

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A307347 Number of 2n-move closed antelope paths on an unbounded chessboard from a given square to the same square.

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A326920 Constant term in the expansion of (-1 + Product_{k=1..n} (1 + x_k + 1/x_k))^n.

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A329075 Constant term in the expansion of ((Sum_{k=-2..2} x^k) * (Sum_{k=-2..2} y^k) - (Sum_{k=-1..1} x^k) * (Sum_{k=-1..1} y^k))^n.

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A329076 Constant term in the expansion of ((Sum_{k=-n..n} x^k) * (Sum_{k=-n..n} y^k) - (Sum_{k=-n+1..n-1} x^k) * (Sum_{k=-n+1..n-1} y^k))^n.

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A329816 Triangular array, read by rows: T(n,k) = [(x*y)^k] (-1 + (1 + x + 1/x)*(1 + y + 1/y))^n for -n <= k <= n.

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A098070 Consider a single king on an infinite chessboard. This sequence gives number of n-move paths when king starting at origin reaches the origin again for the first time at step n.

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A329067 Constant term in the expansion of ((x^5 + x^3 + x + 1/x + 1/x^3 + 1/x^5)(y^5 + y^3 + y + 1/y + 1/y^3 + 1/y^5) - (x^3 + x + 1/x + 1/x^3)(y^3 + y + 1/y + 1/y^3))^(2*n).

A329816 Triangular array, read by rows: T(n,k) = [(xy)^k] (-1 + (1 + x + 1/x)(1 + y + 1/y))^n for -n <= k <= n.