A337390 -id:A337390

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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))).

+10
8

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

FORMULA

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

EXAMPLE

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, ...

MATHEMATICA

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 *)

PROG

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

CROSSREFS

Columns k=0..5 give A000984, A082758, A337390, A245926, A001448, A243946.

Main diagonal gives A337388.

Cf. A337369, A337419.

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Aug 25 2020

STATUS

approved

A337370

Expansion of sqrt(2 / ( (1-12*x+4*x^2) * (1-2*x+sqrt(1-12*x+4*x^2)) )).

+10
4

1, 8, 74, 736, 7606, 80464, 864772, 9400192, 103061158, 1137528688, 12623082284, 140697113792, 1574005263676, 17663830073504, 198760191043784, 2241743315230208, 25335473017856774, 286850379192127664, 3252960763923781276, 36942512756224955456, 420084161646913792724

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Robert Israel, Table of n, a(n) for n = 0..938

FORMULA

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

8*(2*n - 3)*(n - 2)*a(n - 3) - 4*(10*n^2 - 35*n + 27)*a(n - 2) - 2*(10*n^2 + 5*n - 3)*a(n - 1) + (2*n + 1)*n*a(n) = 0. - Robert Israel, Aug 27 2020

a(0) = 1, a(1) = 8 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * (24*n^2-12*n-4) * a(n-1) - 4 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 29 2020

a(n) ~ 2^(n - 5/4) * (1 + sqrt(2))^(2*n + 3/2) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 31 2020

MAPLE

Rec:= 8*(2*n - 3)*(n - 2)*a(n - 3) - 4*(10*n^2 - 35*n + 27)*a(n - 2) - 2*(10*n^2 + 5*n - 3)*a(n - 1) + (2*n + 1)*n*a(n) = 0:

f:= gfun:-rectoproc({Rec, a(0)=1, a(1)=8, a(2)=74}, a(n), remember):

map(f, [$0..30]); # Robert Israel, Aug 27 2020

MATHEMATICA

a[n_] := Sum[2^(n - k) * Binomial[2*k, k] * Binomial[2*n + 1, 2*k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Aug 25 2020 *)

PROG

(PARI) N=40; x='x+O('x^N); Vec(sqrt(2/((1-12*x+4*x^2)*(1-2*x+sqrt(1-12*x+4*x^2)))))

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

CROSSREFS

Column k=2 of A337369.

Cf. A337390.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Aug 25 2020

STATUS

approved

A337421

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI) 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))))

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

CROSSREFS

Column k=2 of A337419.

Cf. A337390.

KEYWORD

sign

AUTHOR

Seiichi Manyama, Aug 27 2020

STATUS

approved

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