A328718 -id:A328718 - cp's OEIS frontend

A329066 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=0..n} x^(2j+1)+1/x^(2j+1)) * (Sum_{j=0..n} y^(2j+1)+1/y^(2j+1)) - (Sum_{j=0..n-1} x^(2j+1)+1/x^(2j+1)) * (Sum_{j=0..n-1} y^(2j+1)+1/y^(2j+1)) )^(2*k).

1, 4, 1, 36, 12, 1, 400, 588, 20, 1, 4900, 49440, 2100, 28, 1, 63504, 5187980, 423440, 4956, 36, 1, 853776, 597027312, 117234740, 1751680, 9540, 44, 1, 11778624, 71962945824, 36938855520, 907687900, 5101200, 16236, 52, 1

Offset: 0

Seiichi Manyama, Nov 03 2019

T(n,k) is the number of (2*k)-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 2*n+1).

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

			Square array begins:
   1,  4,   36,     400,       4900, ...
   1, 12,  588,   49440,    5187980, ...
   1, 20, 2100,  423440,  117234740, ...
   1, 28, 4956, 1751680,  907687900, ...
   1, 36, 9540, 5101200, 4190017860, ...

Seiichi Manyama, Antidiagonals n = 0..50, flattened
Wikipedia, Taxicab geometry.

Columns k=0-1 give A000012, A017113.
Rows n=0-2 give A002894, A329024, A329067.
Main diagonal gives A342964.
Cf. A005408, A328718, A329074, A329078.

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

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

See the second code written in PARI.

A328713 Constant term in the expansion of (1 + x + y + z + 1/x + 1/y + 1/z)^n.

Original entry on oeis.org

1, 1, 7, 19, 127, 511, 3301, 16297, 103279, 570367, 3595177, 21167917, 133789789, 818625133, 5207248879, 32649752779, 209258291599, 1333828204303, 8612806088761, 55546469634733, 361143420408337, 2349709451702737, 15370341546766939, 100695951740818903, 662213750028892429

Offset: 0

Seiichi Manyama, Oct 26 2019

nonn

a(n) is the number of n-step closed walks (from origin to origin) in cubic lattice where each step changes at most one component by -1 or by +1. - Alois P. Heinz, Oct 26 2019

			(1+x+y+z+1/x+1/y+1/z)^2 = x^2 + 1/x^2 + y^2 + 1/y^2 + z^2 + 1/z^2 + 2 * (xy + 1/(xy) + yz + 1/(yz) + zx + 1/(zx) + x/y + y/x + y/z + z/y + z/x + x/z + x + 1/x + y + 1/y + z + 1/z) + 7. So a(2) = 7.

Alois P. Heinz, Table of n, a(n) for n = 0..1189

Row 3 of A328718.

Cf. A002426, A002896, A201805.

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

From Vaclav Kotesovec, Oct 26 2019: (Start)
Recurrence: n^3*a(n) = (2*n - 1)*(2*n^2 - 2*n + 1)*a(n-1) + (n-1)*(34*n^2 - 68*n + 41)*a(n-2) - 38*(n-2)*(n-1)*(2*n - 3)*a(n-3) - 105*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 7^(n + 3/2) / (8 * Pi^(3/2) * n^(3/2)). (End)
E.g.f.: exp(x) * BesselI(0,2*x)^3. - Ilya Gutkovskiy, Oct 26 2019

A328716 Constant term in the expansion of (1 + x_1 + x_2 + ... + x_n + 1/x_1 + 1/x_2 + ... + 1/x_n)^n.

Original entry on oeis.org

1, 1, 5, 19, 217, 1451, 26041, 249705, 6116209, 76432627, 2373097921, 36562658573, 1374991573825, 25188442156333, 1112491608614933, 23620069750701091, 1198207214200181217, 28930659427538020915, 1657461085278025906081, 44848606508761385855085

Offset: 0

Seiichi Manyama, Oct 26 2019

nonn

a(n) is the number of n-step closed walks (from origin to origin) in n-dimensional lattice where each step changes at most one component by -1 or by +1. - Alois P. Heinz, Oct 26 2019

Seiichi Manyama, Table of n, a(n) for n = 0..398 (terms 0..199 from Alois P. Heinz)

Main diagonal of A328718.

a(n) = n! * [x^n] exp(x) * BesselI(0,2*x)^n. - Ilya Gutkovskiy, Oct 26 2019

a(n) ~ c * d^n * n^n, where d = 0.8047104059195202206625458331930618795... and c = 2.12946224998808159475495497... if n is even and c = 1.4189559976544232606562785... if n is odd. - Vaclav Kotesovec, Oct 27 2019

a(7)-a(19) from Alois P. Heinz, Oct 26 2019

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.

A328714 Constant term in the expansion of (1 + w + x + y + z + 1/w + 1/x + 1/y + 1/z)^n.

Original entry on oeis.org

1, 1, 9, 25, 217, 921, 7761, 41889, 345465, 2162617, 17605249, 121120209, 980612161, 7174425025, 58079091513, 442755733065, 3595708057785, 28197440412345, 230133477721665, 1841288167473105, 15113407062476817, 122714906949538257, 1013127345082389513

Offset: 0

Seiichi Manyama, Oct 26 2019

nonn

a(n) is the number of n-step closed walks (from origin to origin) in 4-dimensional lattice where each step changes at most one component by -1 or by +1. - Alois P. Heinz, Oct 26 2019

Alois P. Heinz, Table of n, a(n) for n = 0..1054

Row 4 of A328718.

Cf. A039699.

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

From Vaclav Kotesovec, Oct 26 2019: (Start)
Recurrence: n^4*a(n) = (5*n^4 - 10*n^3 + 10*n^2 - 5*n + 1)*a(n-1) + (n-1)^2*(70*n^2 - 140*n + 113)*a(n-2) - (n-2)*(n-1)*(230*n^2 - 690*n + 583)*a(n-3) - 789*(n-3)*(n-2)^2*(n-1)*a(n-4) + 945*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ 9^(n+2) / (16 * Pi^2 * n^2). (End)
E.g.f.: exp(x) * BesselI(0,2*x)^4. - Ilya Gutkovskiy, Oct 26 2019

a(19)-a(22) from Alois P. Heinz, Oct 26 2019

A328715 Constant term in the expansion of (1 + v + w + x + y + z + 1/v + 1/w + 1/x + 1/y + 1/z)^n.

Original entry on oeis.org

1, 1, 11, 31, 331, 1451, 15101, 85961, 876331, 5917531, 59415961, 450749861, 4481629021, 36869221741, 364723196891, 3177413896031, 31389891383531, 284948206851691, 2818704750978761, 26367817118386661, 261622144605718681, 2502704635436220281, 24932548891897186991

Offset: 0

Seiichi Manyama, Oct 26 2019

nonn

a(n) is the number of n-step closed walks (from origin to origin) in 5-dimensional lattice where each step changes at most one component by -1 or by +1. - Alois P. Heinz, Oct 26 2019

Alois P. Heinz, Table of n, a(n) for n = 0..967

Row 5 of A328718.

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

E.g.f.: exp(x) * BesselI(0,2*x)^5. - Ilya Gutkovskiy, Oct 26 2019
From Vaclav Kotesovec, Oct 27 2019: (Start)
Recurrence: n^5*a(n) = (2*n - 1)*(n^2 - n + 1)*(3*n^2 - 3*n + 1)*a(n-1) + (n-1)*(125*n^4 - 500*n^3 + 903*n^2 - 806*n + 289)*a(n-2) - 2*(n-2)*(n-1)*(2*n - 3)*(135*n^2 - 405*n + 419)*a(n-3) - (n-3)*(n-2)*(n-1)*(3319*n^2 - 13276*n + 14637)*a(n-4) + 3867*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 5)*a(n-5) + 10395*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6).
a(n) ~ 11^(n + 5/2) / (32 * Pi^(5/2) * n^(5/2)). (End)

a(13)-a(22) from Alois P. Heinz, Oct 26 2019

cp's OEIS Frontend

A329066 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=0..n} x^(2*j+1)+1/x^(2*j+1)) * (Sum_{j=0..n} y^(2*j+1)+1/y^(2*j+1)) - (Sum_{j=0..n-1} x^(2*j+1)+1/x^(2*j+1)) * (Sum_{j=0..n-1} y^(2*j+1)+1/y^(2*j+1)) )^(2*k).

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A328713 Constant term in the expansion of (1 + x + y + z + 1/x + 1/y + 1/z)^n.

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A328716 Constant term in the expansion of (1 + x_1 + x_2 + ... + x_n + 1/x_1 + 1/x_2 + ... + 1/x_n)^n.

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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.

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A328714 Constant term in the expansion of (1 + w + x + y + z + 1/w + 1/x + 1/y + 1/z)^n.

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Programs

PARI

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Extensions

A328715 Constant term in the expansion of (1 + v + w + x + y + z + 1/v + 1/w + 1/x + 1/y + 1/z)^n.

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Comments

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Programs

PARI

Formula

Extensions

A329066 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=0..n} x^(2j+1)+1/x^(2j+1)) * (Sum_{j=0..n} y^(2j+1)+1/y^(2j+1)) - (Sum_{j=0..n-1} x^(2j+1)+1/x^(2j+1)) * (Sum_{j=0..n-1} y^(2j+1)+1/y^(2j+1)) )^(2*k).