A333991 -id:A333991 - cp's OEIS frontend

A333990 a(n) = Sum_{k=0..n} n^k * binomial(2n,2k).

1, 2, 17, 208, 3281, 62976, 1419193, 36643328, 1064876737, 34359869440, 1217844546401, 47005113741312, 1961498610274321, 87961440484327424, 4217109422614386761, 215187913985734475776, 11641533109203575871233, 665430291591787349803008, 40065760383961956327231409

Offset: 0

Seiichi Manyama, Sep 04 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..381

Main diagonal of A333988.

Cf. A333991.

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 *)

{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 + sqrt(n))^(2*n)/2 + (1 - sqrt(n))^(2*n)/2.
a(n) ~ exp(2*sqrt(n) - 1) * n^n / 2 * (1 + 2/(3*sqrt(n))). (End)

A333989 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).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, -4, 1, 1, -2, -7, 0, 1, 1, -3, -8, 23, 16, 1, 1, -4, -7, 64, 17, 0, 1, 1, -5, -4, 117, -128, -241, -64, 1, 1, -6, 1, 176, -527, -512, 329, 0, 1, 1, -7, 8, 235, -1264, 237, 4096, 1511, 256, 1, 1, -8, 17, 288, -2399, 3776, 11753, -8192, -5983, 0, 1

Offset: 0

Seiichi Manyama, Sep 04 2020

			Square array begins:
  1,  1,    1,    1,    1,     1, ...
  1,  0,   -1,   -2,   -3,    -4, ...
  1, -4,   -7,   -8,   -7,    -4, ...
  1,  0,   23,   64,  117,   176, ...
  1, 16,   17, -128, -527, -1264, ...
  1,  0, -241, -512,  237,  3776, ...

Main diagonal gives A333991.

Cf. A307884, A333988.

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

cp's OEIS Frontend

A333990 a(n) = Sum_{k=0..n} n^k * binomial(2n,2k).

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A333989 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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cp's OEIS Frontend

A333990 a(n) = Sum_{k=0..n} n^k * binomial(2*n,2*k).

Views

Author

Keywords

Links

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Programs

Mathematica

PARI

Formula

A333989 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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Mathematica

PARI

Formula

A333990 a(n) = Sum_{k=0..n} n^k * binomial(2n,2k).

A333989 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).