A337387 -id:A337387 - cp's OEIS frontend

A337369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of sqrt(2 / ( (1-2(k+4)x+((k-4)x)^2) (1+(k-4)x+sqrt(1-2(k+4)x+((k-4)x)^2)) )).

1, 1, 6, 1, 7, 30, 1, 8, 51, 140, 1, 9, 74, 393, 630, 1, 10, 99, 736, 3139, 2772, 1, 11, 126, 1175, 7606, 25653, 12012, 1, 12, 155, 1716, 14499, 80464, 212941, 51480, 1, 13, 186, 2365, 24310, 183195, 864772, 1787607, 218790, 1, 14, 219, 3128, 37555, 352716, 2351805, 9400192, 15134931, 923780

Offset: 0

Seiichi Manyama, Aug 25 2020

			Square array begins:
     1,     1,     1,      1,      1,      1, ...
     6,     7,     8,      9,     10,     11, ...
    30,    51,    74,     99,    126,    155, ...
   140,   393,   736,   1175,   1716,   2365, ...
   630,  3139,  7606,  14499,  24310,  37555, ...
  2772, 25653, 80464, 183195, 352716, 610897, ...

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

Columns k=0..5 give A002457, A273055, A337370, A245927, A002458, A243947.
Main diagonal gives A337387.
Cf. A337389, A337464.

T[n_, k_] := Sum[If[k == 0, Boole[n == j], k^(n - j)] * Binomial[2*j, j] * Binomial[2*n + 1, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 25 2020 *)

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

T(n,k) = Sum_{j=0..n} k^(n-j) * binomial(2*j,j) * binomial(2*n+1,2*j).
T(0,k) = 1, T(1,k) = k+6 and n * (2*n+1) * (4*n-3) * T(n,k) = (4*n-1) * (4*(k+4)*n^2-2*(k+4)*n-k-2) * T(n-1,k) - (k-4)^2 * (n-1) * (2*n-1) * (4*n+1) * T(n-2,k) for n > 1. - Seiichi Manyama, Aug 29 2020
For fixed k > 0, T(n,k) ~ (2 + sqrt(k))^(2*n + 3/2) / sqrt(8*k*Pi*n). - Vaclav Kotesovec, Aug 31 2020

A337388 a(n) = Sum_{k=0..n} n^(n-k) * binomial(2k,k) binomial(2n,2k).

Original entry on oeis.org

1, 3, 34, 587, 12870, 337877, 10262004, 352436961, 13465074758, 565280386625, 25826066397756, 1274138666796217, 67446164001827356, 3810171540686207283, 228658931521878071080, 14520123059677034441895, 972281769469377542763078, 68443768336740463562683177

Offset: 0

Seiichi Manyama, Aug 25 2020

nonn

Main diagonal of A337389.

Cf. A337387.

a[n_] := Sum[If[n == 0, Boole[n == k], n^(n - k)] * Binomial[2*k, k] * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Aug 25 2020 *)

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

From Vaclav Kotesovec, Aug 31 2020: (Start)
a(n) ~ (2 + sqrt(n))^(2*n + 1/2) / sqrt(8*Pi*n).
a(n) ~ exp(4*sqrt(n) - 4) * n^(n - 1/4) / sqrt(8*Pi) * (1 + 19/(3*sqrt(n)) + 199/(18*n)). (End)

A337465 a(n) = Sum_{k=0..n} (-n)^(n-k) * binomial(2k,k) binomial(2n+1,2k).

Original entry on oeis.org

1, 5, -6, -139, 1654, -5853, -196860, 6258751, -112580442, 985287863, 26443436876, -1897380617625, 72596047613116, -2086036395460171, 39493340495025864, 304974352009838745, -85532651616832374010, 6040114369000387188975, -321378391411642082323524, 14224299551865677212271567

Offset: 0

Seiichi Manyama, Aug 28 2020

sign

Main diagonal of A337464.

Cf. A337387, A337420.

a[n_] := Sum[If[n == n - k == 0, 1, (-n)^(n-k)] * Binomial[2*k, k] * Binomial[2*n+1, 2*k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Apr 29 2021 *)

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

cp's OEIS Frontend

A337369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of sqrt(2 / ( (1-2(k+4)x+((k-4)x)^2) (1+(k-4)x+sqrt(1-2(k+4)x+((k-4)x)^2)) )).

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A337388 a(n) = Sum_{k=0..n} n^(n-k) * binomial(2k,k) binomial(2n,2k).

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Formula

A337465 a(n) = Sum_{k=0..n} (-n)^(n-k) * binomial(2k,k) binomial(2n+1,2k).

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cp's OEIS Frontend

A337369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of sqrt(2 / ( (1-2*(k+4)*x+((k-4)*x)^2) * (1+(k-4)*x+sqrt(1-2*(k+4)*x+((k-4)*x)^2)) )).

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A337388 a(n) = Sum_{k=0..n} n^(n-k) * binomial(2*k,k) * binomial(2*n,2*k).

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Formula

A337465 a(n) = Sum_{k=0..n} (-n)^(n-k) * binomial(2*k,k) * binomial(2*n+1,2*k).

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A337369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of sqrt(2 / ( (1-2(k+4)x+((k-4)x)^2) (1+(k-4)x+sqrt(1-2(k+4)x+((k-4)x)^2)) )).

A337388 a(n) = Sum_{k=0..n} n^(n-k) * binomial(2k,k) binomial(2n,2k).

A337465 a(n) = Sum_{k=0..n} (-n)^(n-k) * binomial(2k,k) binomial(2n+1,2k).