A337419 -id:A337419 - cp's OEIS frontend

A337389 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, 3, 6, 1, 4, 19, 20, 1, 5, 34, 141, 70, 1, 6, 51, 328, 1107, 252, 1, 7, 70, 587, 3334, 8953, 924, 1, 8, 91, 924, 7123, 34904, 73789, 3432, 1, 9, 114, 1345, 12870, 89055, 372436, 616227, 12870, 1, 10, 139, 1856, 20995, 184756, 1135005, 4027216, 5196627, 48620

Offset: 0

Seiichi Manyama, Aug 25 2020

			Square array begins:
    1,    1,     1,     1,      1,      1, ...
    2,    3,     4,     5,      6,      7, ...
    6,   19,    34,    51,     70,     91, ...
   20,  141,   328,   587,    924,   1345, ...
   70, 1107,  3334,  7123,  12870,  20995, ...
  252, 8953, 34904, 89055, 184756, 337877, ...

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

Columns k=0..5 give A000984, A082758, A337390, A245926, A001448, A243946.
Main diagonal gives A337388.
Cf. A337369, A337419.

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 25 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) = k+2 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
For fixed k > 0, T(n,k) ~ (2 + sqrt(k))^(2*n + 1/2) / sqrt(8*Pi*n). - Vaclav Kotesovec, Aug 31 2020
Conjecture: the k-th column entries, k >= 0, are given by [x^n] ( (1 + (k-2)*x + x^2)*(1 + x)^2/(1 - x)^2 )^n. This is true for k = 0 and k = 4. - Peter Bala, May 03 2022

A337464 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, 5, 30, 1, 4, 11, 140, 1, 3, -6, -29, 630, 1, 2, -21, -120, -365, 2772, 1, 1, -34, -139, -266, -1409, 12012, 1, 0, -45, -92, 531, 2520, -155, 51480, 1, -1, -54, 15, 1654, 6489, 17380, 29485, 218790, 1, -2, -61, 176, 2755, 4828, -9723, -13104, 170035, 923780

Offset: 0

Seiichi Manyama, Aug 28 2020

			Square array begins:
     1,     1,    1,    1,    1,     1, ...
     6,     5,    4,    3,    2,     1, ...
    30,    11,   -6,  -21,  -34,   -45, ...
   140,   -29, -120, -139,  -92,    15, ...
   630,  -365, -266,  531, 1654,  2755, ...
  2772, -1409, 2520, 6489, 4828, -5853, ...

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

Columns k=0..4 give A002457, A337394, A337466, A337467, A337397.
Main diagonal gives A337465.
Cf. A337369, A337419.

T[n_, k_] := Sum[If[k == n-j == 0, 1, (-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, Apr 29 2021 *)

{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) = 6-k 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

A337421 Expansion of sqrt((1-6x+sqrt(1-4x+36x^2)) / (2 (1-4x+36x^2))).

Original entry on oeis.org

1, 0, -14, -48, 198, 2080, 1780, -57120, -270522, 796992, 11771676, 18981600, -314843364, -1841666112, 3400749352, 74960197312, 175979793990, -1853840247168, -13190663057780, 11783856595680, 496784970525748, 1536657455021760, -11053154849810472, -96149956882617792, 4480143410034972

Offset: 0

Seiichi Manyama, Aug 27 2020

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

Column k=2 of A337419.

Cf. A337390.

a[n_] := Sum[(-2)^(n - k) * Binomial[2*k, k] * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 25, 0] (* Amiram Eldar, Aug 27 2020 *)

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

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

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

a(0) = 1, a(1) = 0 and n * (2*n-1) * (4*n-5) * a(n) = (4*n-3) * (8*n^2-12*n+4) * a(n-1) - 36 * (n-1) * (2*n-3) * (4*n-1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 28 2020

A337422 Expansion of sqrt((1-7x+sqrt(1-2x+49x^2)) / (2 (1-2x+49x^2))).

Original entry on oeis.org

1, -1, -21, -7, 739, 1629, -26859, -118329, 922419, 6886397, -27414191, -358533429, 539620621, 17229485987, 8782716411, -769962297447, -1897237412973, 31786556599917, 149610560086113, -1182765435388341, -9268347520205991, 37049669347266471, 505738623506722431

Offset: 0

Seiichi Manyama, Aug 27 2020

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

Column k=3 of A337419.

Cf. A245926.

a[n_] := Sum[(-3)^(n - k) * Binomial[2*k, k] * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 23, 0] (* Amiram Eldar, Aug 27 2020 *)

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

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

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

a(0) = 1, a(1) = -1 and n * (2*n-1) * (4*n-5) * a(n) = (4*n-3) * (4*n^2-6*n+3) * a(n-1) - 49 * (n-1) * (2*n-3) * (4*n-1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 28 2020

A337393 Expansion of sqrt((1-5x+sqrt(1-6x+25x^2)) / (2 (1-6x+25x^2))).

Original entry on oeis.org

1, 1, -5, -41, -125, 131, 3301, 15625, 16115, -254525, -1813055, -4617755, 14903725, 192390589, 767919595, -28588201, -18144634861, -105011253485, -184605603311, 1406589226405, 12610893954745, 40402054036345, -63847551719825, -1340432504352485, -6346702151685475

Offset: 0

Seiichi Manyama, Aug 25 2020

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

Column k=1 of A337419.

Cf. A082758, A188599, A337394.

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

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

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

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

a(0) = 1, a(1) = 1 and n * (2*n-1) * (4*n-5) * a(n) = (4*n-3) * (12*n^2-18*n+5) * a(n-1) - 25 * (n-1) * (2*n-3) * (4*n-1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 28 2020

A337396 Expansion of sqrt((1-8x+sqrt(1+64x^2)) / (2 * (1+64*x^2))).

Original entry on oeis.org

1, -2, -26, 76, 1222, -3772, -64676, 203992, 3607622, -11510636, -207302156, 666187432, 12142184476, -39211413464, -720760216328, 2335857124016, 43208062233158, -140406756766796, -2609918906614652, 8498967890177416, 158596941629422132, -517334728427373704, -9684521991498517112

Offset: 0

Seiichi Manyama, Aug 26 2020

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

Column k=4 of A337419.

Cf. A001448, A193619, A337397.

a[n_] := Sum[(-4)^(n - k) * Binomial[2*k, k] * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 23, 0] (* Amiram Eldar, Aug 26 2020 *)

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

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

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

a(0) = 1, a(1) = -2 and n * (2*n-1) * (4*n-5) * a(n) = (4*n-3) * 2 * a(n-1) - 64 * (n-1) * (2*n-3) * (4*n-1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 28 2020

A337420 a(n) = Sum_{k=0..n} (-n)^(n-k) * binomial(2k,k) binomial(2n,2k).

Original entry on oeis.org

1, 1, -14, -7, 1222, -14873, 26196, 3522955, -110841786, 2088947819, -15869398244, -823790768205, 55262757020956, -2199333670723343, 65894251730104552, -1235877788883794355, -18904175519674543546, 3743957841955101437667, -268850524243738610546292, 14826380281246309472525851

Offset: 0

Seiichi Manyama, Aug 27 2020

sign

Main diagonal of A337419.

Cf. A337388.

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, 20, 0] (* Amiram Eldar, Aug 27 2020 *)

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

cp's OEIS Frontend

A337389 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))).

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A337464 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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A337421 Expansion of sqrt((1-6*x+sqrt(1-4*x+36*x^2)) / (2 * (1-4*x+36*x^2))).

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A337422 Expansion of sqrt((1-7*x+sqrt(1-2*x+49*x^2)) / (2 * (1-2*x+49*x^2))).

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A337393 Expansion of sqrt((1-5*x+sqrt(1-6*x+25*x^2)) / (2 * (1-6*x+25*x^2))).

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A337396 Expansion of sqrt((1-8*x+sqrt(1+64*x^2)) / (2 * (1+64*x^2))).

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A337420 a(n) = Sum_{k=0..n} (-n)^(n-k) * binomial(2*k,k) * binomial(2*n,2*k).

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A337389 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))).

A337464 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)) )).

A337421 Expansion of sqrt((1-6x+sqrt(1-4x+36x^2)) / (2 (1-4x+36x^2))).

A337422 Expansion of sqrt((1-7x+sqrt(1-2x+49x^2)) / (2 (1-2x+49x^2))).

A337393 Expansion of sqrt((1-5x+sqrt(1-6x+25x^2)) / (2 (1-6x+25x^2))).

A337396 Expansion of sqrt((1-8x+sqrt(1+64x^2)) / (2 * (1+64*x^2))).

A337420 a(n) = Sum_{k=0..n} (-n)^(n-k) * binomial(2k,k) binomial(2n,2k).