A337464 -id:A337464 - cp's OEIS frontend

A337419 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of sqrt((1-(k+4)x+sqrt(1+2(k-4)x+((k+4)x)^2)) / (2 * (1+2(k-4)x+((k+4)*x)^2))).

1, 1, 2, 1, 1, 6, 1, 0, -5, 20, 1, -1, -14, -41, 70, 1, -2, -21, -48, -125, 252, 1, -3, -26, -7, 198, 131, 924, 1, -4, -29, 76, 739, 2080, 3301, 3432, 1, -5, -30, 195, 1222, 1629, 1780, 15625, 12870, 1, -6, -29, 344, 1395, -3772, -26859, -57120, 16115, 48620

Offset: 0

Seiichi Manyama, Aug 27 2020

			Square array begins:
    1,    1,    1,    1,     1,      1, ...
    2,    1,    0,   -1,    -2,     -3, ...
    6,   -5,  -14,  -21,   -26,    -29, ...
   20,  -41,  -48,   -7,    76,    195, ...
   70, -125,  198,  739,  1222,   1395, ...
  252,  131, 2080, 1629, -3772, -14873, ...

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

Columns k=0..4 give A000984, A337393, A337421, A337422, A337396.
Main diagonal gives A337420.
Cf. A337389, A337464.

T[n_, k_] := Sum[If[k == 0, Boole[n == j], (-k)^(n - j)] * Binomial[2*j, j] * Binomial[2*n, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 27 2020 *)

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

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

T(0,k) = 1, T(1,k) = 2-k and n * (2*n-1) * (4*n-5) * T(n,k) = (4*n-3) * (-4*(k-4)*n^2+6*(k-4)*n-k+6) * T(n-1,k) - (k+4)^2 * (n-1) * (2*n-3) * (4*n-1) * T(n-2,k) for n > 1. - Seiichi Manyama, Aug 28 2020

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)) )).

Original entry on oeis.org

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

A337394 Expansion of sqrt(2 / ( (1-6x+25x^2) * (1-5x+sqrt(1-6x+25*x^2)) )).

Original entry on oeis.org

1, 5, 11, -29, -365, -1409, -155, 29485, 170035, 309775, -2064655, -18909175, -61552739, 81290561, 1901796395, 9145986419, 8604744275, -165227713249, -1168032362879, -2913302013175, 10702975797545, 132134872338925, 519716440255535, -109051949915065, -13098011769247075

Offset: 0

Seiichi Manyama, Aug 25 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=1 of A337464.

Cf. A188599, A273055, A337393.

a[n_] := Sum[(-1)^(n-k) * Binomial[2*k, k] * Binomial[2*n+1, 2*k], {k, 0, n}]; Array[a, 25, 0] (* Amiram Eldar, Apr 29 2021 *)

N=40; x='x+O('x^N); Vec(sqrt(2/((1-6*x+25*x^2)*(1-5*x+sqrt(1-6*x+25*x^2)))))

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

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

a(0) = 1, a(1) = 5 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * (12*n^2-6*n-1) * a(n-1) - 25 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 29 2020

A337397 Expansion of sqrt(2 / ( (1+64x^2) (1-8x+sqrt(1+64x^2)) )).

Original entry on oeis.org

1, 2, -34, -92, 1654, 4828, -88724, -268088, 4984486, 15361708, -287691196, -898052872, 16901635516, 53234639768, -1005474931816, -3187958034544, 60375963282182, 192405594166988, -3651655920615596, -11684176213422568, 222132094724096852, 713091439789994824, -13575872676384218776

Offset: 0

Seiichi Manyama, Aug 26 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=4 of A337464.

Cf. A002458, A193618, A337396.

a[n_] := Sum[(-4)^(n - k) * Binomial[2*k, k] * Binomial[2*n + 1, 2*k], {k, 0, n}]; Array[a, 23, 0] (* Amiram Eldar, Aug 26 2020 *)
CoefficientList[Series[Sqrt[2/((1+64x^2)(1-8x+Sqrt[1+64x^2]))],{x,0,30}],x] (* Harvey P. Dale, Jul 24 2021 *)

N=40; x='x+O('x^N); Vec(sqrt(2/((1+64*x^2)*(1-8*x+sqrt(1+64*x^2)))))

{a(n) = sum(k=0, n, (-4)^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))}

a(n) = Sum_{k=0..n} (-4)^(n-k) * binomial(2*k,k) * binomial(2*n+1,2*k).

a(0) = 1, a(1) = 2 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * 2 * a(n-1) - 64 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 29 2020

A337466 Expansion of sqrt(2 / ( (1-4x+36x^2) * (1-6x+sqrt(1-4x+36*x^2)) )).

Original entry on oeis.org

1, 4, -6, -120, -266, 2520, 17380, -13104, -599130, -1853544, 12391116, 108252144, 6439356, -3577917200, -14043012984, 65962248352, 730407220998, 602517029400, -22507424996420, -108316306187600, 347406564086868, 5073542740156752, 7904100039294456, -143838603813578400

Offset: 0

Seiichi Manyama, Aug 28 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=2 of A337464.

Cf. A337370, A337421.

a[n_] := Sum[(-2)^(n-k) * Binomial[2*k, k] * Binomial[2*n+1, 2*k], {k, 0, n}]; Array[a, 24, 0] (* Amiram Eldar, Apr 29 2021 *)
CoefficientList[Series[Sqrt[2/((1-4x+36x^2)(1-6x+Sqrt[1-4x+36x^2]))],{x,0,40}],x] (* Harvey P. Dale, Sep 07 2023 *)

N=40; x='x+O('x^N); Vec(sqrt(2/((1-4*x+36*x^2)*(1-6*x+sqrt(1-4*x+36*x^2)))))

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

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

a(0) = 1, a(1) = 4 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * (8*n^2-4*n) * a(n-1) - 36 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 29 2020

A337467 Expansion of sqrt(2 / ( (1-2x+49x^2) * (1-7x+sqrt(1-2x+49*x^2)) )).

Original entry on oeis.org

1, 3, -21, -139, 531, 6489, -9723, -292293, -135117, 12514313, 29905809, -501239553, -2310673379, 18245192679, 140574917259, -562805403867, -7557237645741, 11275709877369, 371974318253601, 201852054629631, -16932135947326551, -42530838930147813, 709138646702505999

Offset: 0

Seiichi Manyama, Aug 28 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=3 of A337464.

Cf. A245927, A337422.

a[n_] := Sum[(-3)^(n-k) * Binomial[2*k, k] * Binomial[2*n+1, 2*k], {k, 0, n}]; Array[a, 23, 0] (* Amiram Eldar, Apr 29 2021 *)

N=40; x='x+O('x^N); Vec(sqrt(2/((1-2*x+49*x^2)*(1-7*x+sqrt(1-2*x+49*x^2)))))

{a(n) = sum(k=0, n, (-3)^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))}

a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*k,k) * binomial(2*n+1,2*k).

a(0) = 1, a(1) = 3 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * (4*n^2-2*n+1) * a(n-1) - 49 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 29 2020

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

A337419 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of sqrt((1-(k+4)*x+sqrt(1+2*(k-4)*x+((k+4)*x)^2)) / (2 * (1+2*(k-4)*x+((k+4)*x)^2))).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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)) )).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A337394 Expansion of sqrt(2 / ( (1-6*x+25*x^2) * (1-5*x+sqrt(1-6*x+25*x^2)) )).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A337397 Expansion of sqrt(2 / ( (1+64*x^2) * (1-8*x+sqrt(1+64*x^2)) )).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A337466 Expansion of sqrt(2 / ( (1-4*x+36*x^2) * (1-6*x+sqrt(1-4*x+36*x^2)) )).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A337467 Expansion of sqrt(2 / ( (1-2*x+49*x^2) * (1-7*x+sqrt(1-2*x+49*x^2)) )).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

A337419 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of sqrt((1-(k+4)x+sqrt(1+2(k-4)x+((k+4)x)^2)) / (2 * (1+2(k-4)x+((k+4)*x)^2))).

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)) )).

A337394 Expansion of sqrt(2 / ( (1-6x+25x^2) * (1-5x+sqrt(1-6x+25*x^2)) )).

A337397 Expansion of sqrt(2 / ( (1+64x^2) (1-8x+sqrt(1+64x^2)) )).

A337466 Expansion of sqrt(2 / ( (1-4x+36x^2) * (1-6x+sqrt(1-4x+36*x^2)) )).

A337467 Expansion of sqrt(2 / ( (1-2x+49x^2) * (1-7x+sqrt(1-2x+49*x^2)) )).

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