A377239 Expansion of sqrt(1 + 4*x/sqrt(1 - 4*x)).

1, 2, 2, 8, 22, 80, 268, 976, 3494, 12896, 47556, 177744, 666236, 2515104, 9525976, 36240032, 138269958, 529193152, 2030242708, 7807266512, 30082317460, 116126074400, 449014896040, 1738805527520, 6742758295900, 26180338089280, 101769629532968, 396031222808096, 1542673037902904

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1657

Formula

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

a(n) ~ 2^(2*n) / (Gamma(1/4) * n^(3/4)) * (1 - Gamma(1/4)^2 / (Pi*2^(7/2)*sqrt(n))). - Vaclav Kotesovec, May 03 2025

D-finite with recurrence: 32*(4*n - 1)*(4*n + 1)*a(n) - 48*(8*n^2 + 20*n + 11)*a(n + 1) + 24*(7 + 4*n)*a(n + 2) + 4*(7*n + 20)*(2*n + 5)*a(n + 3) - 2*(n + 4)*(7*n + 23)*a(n + 4) + (n + 5)*(n + 4)*a(n + 5) = 0. - Robert Israel, Feb 24 2026

Maple

f:= rectoproc({32*(4*n - 1)*(4*n + 1)*a(n) - 48*(8*n^2 + 20*n + 11)*a(n + 1) + 24*(7 + 4*n)*a(n + 2) + 4*(7*n + 20)*(2*n + 5)*a(n + 3) - 2*(n + 4)*(7*n + 23)*a(n + 4) + (n + 5)*(n + 4)*a(n + 5), a(0) = 1, a(1) = 2, a(2) = 2, a(3) = 8, a(4) = 22}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Feb 24 2026

Prog

(PARI) a(n) = 4^n*sum(k=0, n, binomial(1/2, k)*binomial(n-k/2-1, n-k));

Crossrefs

Cf. A377240.

Sequence in context: A130102 A151384 A300460 * A290613 A113464 A353252

Adjacent sequences: A377236 A377237 A377238 * A377240 A377241 A377242

Keyword

nonn

Author

Seiichi Manyama, Oct 21 2024

Status

approved