A360309 - OEIS

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A360309

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

1, 0, 0, 2, 2, 2, 8, 14, 20, 46, 92, 158, 314, 630, 1176, 2274, 4498, 8674, 16804, 32990, 64358, 125414, 245832, 481674, 942912, 1850122, 3633220, 7133730, 14020694, 27578954, 54261912, 106819006, 210411028, 414619486, 817344908, 1611978734, 3180333830, 6276743430

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OFFSET

0,4

LINKS

Table of n, a(n) for n=0..37.

FORMULA

G.f.: 1 / sqrt(1-4*x^3/(1-x)).

n*a(n) = 2*(n-1)*a(n-1) - (n-2)*a(n-2) + 2*(2*n-3)*a(n-3) - 2*(2*n-6)*a(n-4).

a(n) ~ 2^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 18 2023

PROG

(PARI) a(n) = sum(k=0, n\3, binomial(n-1-2*k, n-3*k)*binomial(2*k, k));

(PARI) my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x^3/(1-x)))