A333989 -id:A333989 - cp's OEIS frontend

A333988 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1-(k+1)x) / (1-2(k+1)x+((k-1)x)^2).

1, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 4, 17, 32, 1, 1, 5, 28, 99, 128, 1, 1, 6, 41, 208, 577, 512, 1, 1, 7, 56, 365, 1552, 3363, 2048, 1, 1, 8, 73, 576, 3281, 11584, 19601, 8192, 1, 1, 9, 92, 847, 6016, 29525, 86464, 114243, 32768, 1, 1, 10, 113, 1184, 10033, 62976, 265721, 645376, 665857, 131072, 1

Offset: 0

Seiichi Manyama, Sep 04 2020

			Square array begins:
  1,   1,    1,     1,     1,     1, ...
  1,   2,    3,     4,     5,     6, ...
  1,   8,   17,    28,    41,    56, ...
  1,  32,   99,   208,   365,   576, ...
  1, 128,  577,  1552,  3281,  6016, ...
  1, 512, 3363, 11584, 29525, 62976, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Column k=0..9 give A000012, A081294, A001541, A090965, A083884, A099140, A099141, A099142, A165224, A026244.
Main diagonal gives A333990.
Cf. A009999, A307883, A337389, A333989.

T[n_, 0] := 1; T[n_, k_] := Sum[k^j * Binomial[2*n, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Sep 04 2020 *)

{T(n, k) = sum(j=0, n, k^j*binomial(2*n, 2*j))}

T(n,k) = Sum_{j=0..n} k^j * binomial(2*n,2*j).

T(0,k) = 1, T(1,k) = k+1 and T(n,k) = 2 * (k+1) * T(n-1,k) - (k-1)^2 * T(n-2,k) for n>1.

A333991 a(n) = Sum_{k=0..n} (-n)^k * binomial(2n,2k).

Original entry on oeis.org

1, 0, -7, 64, -527, 3776, -7199, -712704, 28545857, -881543168, 25615822601, -733594255360, 20859188600881, -580152163418112, 15048530008948913, -311489672222081024, 713562283940993281, 511135051171610230784, -48010258775057340355559, 3439412411849176925601792

Offset: 0

Seiichi Manyama, Sep 04 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..386

Main diagonal of A333989.

Cf. A065440, A333990.

a[0] = 1; a[n_] := Sum[(-n)^k * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Sep 04 2020 *)
Table[Hypergeometric2F1[1/2 - n, -n, 1/2, -n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 05 2020 *)
Table[Cos[2*n*ArcTan[Sqrt[n]]] * (n + 1)^n, {n, 0, 20}] // Round (* Vaclav Kotesovec, Sep 05 2020 *)

{a(n) = sum(k=0, n, (-n)^k*binomial(2*n, 2*k))}

From Vaclav Kotesovec, Sep 05 2020: (Start)
a(n) = hypergeometric2F1(1/2 - n, -n, 1/2, -n).
a(n) = (1 + i*sqrt(n))^(2*n)/2 + (1 - i*sqrt(n))^(2*n)/2, where i is the imaginary unit.
a(n) = cos(2*n*arctan(sqrt(n))) * (n + 1)^n. (End)

cp's OEIS Frontend

A333988 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1-(k+1)x) / (1-2(k+1)x+((k-1)x)^2).

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A333991 a(n) = Sum_{k=0..n} (-n)^k * binomial(2n,2k).

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cp's OEIS Frontend

A333988 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1-(k+1)*x) / (1-2*(k+1)*x+((k-1)*x)^2).

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A333991 a(n) = Sum_{k=0..n} (-n)^k * binomial(2*n,2*k).

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A333988 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1-(k+1)x) / (1-2(k+1)x+((k-1)x)^2).

A333991 a(n) = Sum_{k=0..n} (-n)^k * binomial(2n,2k).