A377241 Expansion of 1/sqrt(1 - 4*x*sqrt(1 + 4*x)).

1, 2, 10, 40, 198, 912, 4540, 22032, 110598, 550240, 2783220, 14041104, 71469468, 363722912, 1860732856, 9526388000, 48933655494, 251617706688, 1296741709252, 6690068720016, 34571534145588, 178829630116896, 926198946642760, 4801277360678240, 24913966227624604

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1376

Formula

a(n) = (-4)^n * Sum_{k=0..n} binomial(-1/2,k) * binomial(n-3*k/2-1,n-k).

D-finite with recurrence: 96*(4*n + 9)*(4*n + 3)*a(n) + 16*(64*n^2 + 288*n + 305)*a(n + 1) + 8*(2*n + 7)*(14*n + 37)*a(n + 2) - 4*(2*n^2 - 5*n - 36)*a(n + 3) - 2*(n + 4)*(5*n + 13)*a(n + 4) - (n + 5)*(n + 4)*a(n + 5) = 0.

Maple

f:= gfun:-rectoproc({96*(4*n + 9)*(4*n + 3)*a(n) + 16*(64*n^2 + 288*n + 305)*a(n + 1) + 8*(2*n + 7)*(14*n + 37)*a(n + 2) - 4*(2*n^2 - 5*n - 36)*a(n + 3) - 2*(n + 4)*(5*n + 13)*a(n + 4) - (n + 5)*(n + 4)*a(n + 5), a(0) = 1, a(1) = 2, a(2) = 10, a(3) = 40, a(4) = 198}, 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-3*k/2-1, n-k));

Crossrefs

Cf. A377242.

Sequence in context: A151026 A151027 A188325 * A151028 A282619 A080252

Adjacent sequences: A377238 A377239 A377240 * A377242 A377243 A377244

Keyword

nonn

Author

Seiichi Manyama, Oct 21 2024

Status

approved