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A337390 Expansion of sqrt((1-2*x+sqrt(1-12*x+4*x^2)) / (2 * (1-12*x+4*x^2))). 4
1, 4, 34, 328, 3334, 34904, 372436, 4027216, 43976774, 483860632, 5355697084, 59569288816, 665238165916, 7454247891952, 83769667651816, 943744775565728, 10655369806377542, 120535523282756632, 1365840013196530348, 15500428304345011504, 176148760580561346484 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..939

FORMULA

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

a(0) = 1, a(1) = 4 and n * (2*n-1) * (4*n-5) * a(n) = (4*n-3) * (24*n^2-36*n+8) * a(n-1) - 4 * (n-1) * (2*n-3) * (4*n-1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 28 2020

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

MATHEMATICA

a[n_] := Sum[2^(n - k) * Binomial[2*k, k] * Binomial[2*n, 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((1-2*x+sqrt(1-12*x+4*x^2))/(2*(1-12*x+4*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 A337389.

Cf. A337370, A337421.

Sequence in context: A005569 A232910 A208215 * A025572 A093137 A332617

Adjacent sequences:  A337387 A337388 A337389 * A337391 A337392 A337393

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Aug 25 2020

STATUS

approved

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Last modified May 16 13:42 EDT 2021. Contains 343947 sequences. (Running on oeis4.)