A328874 -id:A328874 - cp's OEIS frontend

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.

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

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

A329819 Triangular array, read by rows: T(n,k) = [(xyz)^k] (-1 + (1 + x + 1/x)(1 + y + 1/y)(1 + z + 1/z))^n for -n <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 6, 26, 6, 1, 1, 24, 195, 264, 195, 24, 1, 1, 60, 898, 3276, 5646, 3276, 898, 60, 1, 1, 120, 3065, 22260, 72730, 101520, 72730, 22260, 3065, 120, 1, 1, 210, 8526, 105690, 581475, 1510860, 2103740, 1510860, 581475, 105690, 8526, 210, 1

Offset: 0

Seiichi Manyama, Nov 21 2019

			Triangle begins:
                                 1;
                         1,      0,     1;
                  1,     6,     26,     6,     1;
           1,    24,   195,    264,   195,    24,    1;
     1,   60,   898,  3276,   5646,  3276,   898,   60,   1;
1, 120, 3065, 22260, 72730, 101520, 72730, 22260, 3065, 120, 1;

Seiichi Manyama, Rows n = 0..25, flattened

T(n,0) gives A328874.

Cf. A260492, A329816, A329820.

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

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

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.

A384105 Triangle read by rows: T(n,k) is the number of binary relations on a set of n objects, exactly k of which are self referencing, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 3, 4, 3, 16, 36, 36, 16, 218, 752, 1104, 752, 218, 9608, 45960, 90416, 90416, 45960, 9608, 1540944, 9133760, 22692704, 30194176, 22692704, 9133760, 1540944, 882033440, 6154473664, 18425858880, 30679088480, 30679088480, 18425858880, 6154473664, 882033440

Offset: 0

Peter Dolland, May 19 2025

Also the number of essentially different simple digraphs on a node set A of size n with a distinguished subset B of size k, where elements are indistinguishable within B and within A \ B.

			Triangle starts:
            1
            1,              1
            3,              4,              3
           16,             36,             36,              16
          218,            752,           1104,             752,             218
         9608,          45960,          90416,           90416,           45960, ...
      1540944,        9133760,       22692704,        30194176,        22692704, ...
    882033440,     6154473664,    18425858880,     30679088480,     30679088480, ...
1793359192848, 14334221970688, 50138592081152, 100240050239744, 125284653092864, ...
...

Peter Dolland, Python calculation of T(n,k)

Cf. A000273 (edge cases), A000595 (row sums), A353996, A328874, A383617.

T(n,k) = T(n,n-k).
T(n,0) = T(n,n) = A000273(n).
T(n,1) = T(n,n-1) = A353996(n+1) = A329874(n,4).
Sum_{k=0..n} T(n,k) = A000595(n).

cp's OEIS Frontend

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.

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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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A329819 Triangular array, read by rows: T(n,k) = [(x*y*z)^k] (-1 + (1 + x + 1/x)*(1 + y + 1/y)*(1 + z + 1/z))^n for -n <= k <= 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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A384105 Triangle read by rows: T(n,k) is the number of binary relations on a set of n objects, exactly k of which are self referencing, 0 <= k <= n.

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Crossrefs

Formula

A329819 Triangular array, read by rows: T(n,k) = [(xyz)^k] (-1 + (1 + x + 1/x)(1 + y + 1/y)(1 + z + 1/z))^n for -n <= k <= n.