A328813 -id:A328813 - cp's OEIS frontend

A328747 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-1)^(n-i)binomial(n,i)Sum_{j=0..i} binomial(i,j)^k.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 7, 7, 1, 0, 1, 1, 15, 31, 19, 1, 0, 1, 1, 31, 115, 175, 51, 1, 0, 1, 1, 63, 391, 1255, 991, 141, 1, 0, 1, 1, 127, 1267, 8071, 13671, 5881, 393, 1, 0, 1, 1, 255, 3991, 49399, 161671, 160461, 35617, 1107, 1, 0

Offset: 0

Seiichi Manyama, Oct 27 2019

T(n,k) is the constant term in the expansion of (-1 + Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0.

For fixed k > 0, T(n,k) ~ (2^k - 1)^(n + (k-1)/2) / (2^((k-1)^2/2) * sqrt(k) * (Pi*n)^((k-1)/2)). - Vaclav Kotesovec, Oct 28 2019

			Square array begins:
   1, 1,  1,   1,     1,      1, ...
   1, 1,  1,   1,     1,      1, ...
   0, 1,  3,   7,    15,     31, ...
   0, 1,  7,  31,   115,    391, ...
   0, 1, 19, 175,  1255,   8071, ...
   0, 1, 51, 991, 13671, 161671, ...

Seiichi Manyama, Antidiagonals n = 0..100, flattened

Columns k=0..5 give A019590(n+1), A000012, A002426, A172634, A328725, A328750.
Main diagonal gives A328811.
T(n,n+1) gives A328813.
Cf. A309010, A328748, A328807.

T[n_, k_] := Sum[(-1)^(n-i) * Binomial[n, i] * Sum[Binomial[i, j]^k, {j, 0, i}], {i, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 06 2021 *)

A328812 Constant term in the expansion of (Product_{k=1..n} (1 + x_k) + Product_{k=1..n} (1 + 1/x_k))^n.

Original entry on oeis.org

1, 2, 10, 164, 9826, 2031252, 1622278624, 4579408029576, 51207103076632066, 2052124795850957537060, 330463219813679264204224300, 192454957455454582636391397662856, 454577215426865313388106323928590128736, 3907905904547764847197154889183844343802986600

Offset: 0

Seiichi Manyama, Oct 28 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..58

Cf. A167010, A309010, A328813, A328814.

a[n_] := Sum[Binomial[n, k]^(n + 1), {k, 0, n}]; Array[a, 14, 0] (* Amiram Eldar, May 06 2021 *)

{a(n) = sum(k=0, n, binomial(n, k)^(n+1))}

a(n) = A309010(n,n+1) = Sum_{k=0..n} binomial(n,k)^(n+1).

a(n) ~ c * exp(-1/4) * 2^((2*n+1)*(n+1)/2) / (Pi*n)^((n+1)/2), where c = A218792 = Sum_{k = -infinity..infinity} exp(-2*k^2) = 1.271341522189... and c = Sum_{k = -infinity..infinity} exp(-2*(k+1/2)^2) = 1.23528676585389... if n is odd. - Vaclav Kotesovec, May 06 2021

A328814 Constant term in the expansion of (-2 + Product_{k=1..n} (1 + x_k) + Product_{k=1..n} (1 + 1/x_k))^n.

Original entry on oeis.org

1, 0, 6, 72, 6690, 1536000, 1398496680, 4165565871600, 48724656010825410, 1991141239554487077120, 325362786100184356140612996, 190695111051826003327799496771600, 452459020719698368348441955010421696800

Offset: 0

Seiichi Manyama, Oct 28 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..58

Cf. A328748, A328812, A328813.

a[n_] := Sum[(-2)^(n-i) * Binomial[n, i] * Sum[Binomial[i, j]^(n+1), {j, 0, i}], {i, 0, n}]; Array[a, 13, 0] (* Amiram Eldar, May 06 2021 *)

{a(n) = sum(i=0, n, (-2)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^(n+1)))}

a(n) = A328748(n,n+1) = Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^(n+1).

cp's OEIS Frontend

A328747 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-1)^(n-i)binomial(n,i)Sum_{j=0..i} binomial(i,j)^k.

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A328812 Constant term in the expansion of (Product_{k=1..n} (1 + x_k) + Product_{k=1..n} (1 + 1/x_k))^n.

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A328814 Constant term in the expansion of (-2 + Product_{k=1..n} (1 + x_k) + Product_{k=1..n} (1 + 1/x_k))^n.

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cp's OEIS Frontend

A328747 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

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Mathematica

A328812 Constant term in the expansion of (Product_{k=1..n} (1 + x_k) + Product_{k=1..n} (1 + 1/x_k))^n.

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A328814 Constant term in the expansion of (-2 + Product_{k=1..n} (1 + x_k) + Product_{k=1..n} (1 + 1/x_k))^n.

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A328747 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-1)^(n-i)binomial(n,i)Sum_{j=0..i} binomial(i,j)^k.