A376574 G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)/(1 - x^3)).
1, 1, 2, 5, 15, 46, 147, 486, 1646, 5684, 19940, 70864, 254592, 923153, 3374046, 12417246, 45975677, 171141378, 640105278, 2404375805, 9066188052, 34305301482, 130219435385, 495735347502, 1892254721982, 7240580768021, 27768359445128, 106718055778871
Offset: 0
Keywords
Programs
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Maple
A376574 := proc(n) add(A000108(n-3*k)*binomial(n-2*k-1,k),k=0..floor(n/3)) ; end proc: seq(A376574(n),n=0..80) ; # R. J. Mathar, Oct 24 2024
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PARI
a(n) = sum(k=0, n\3, binomial(n-2*k-1, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));
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PARI
my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x/(1-x^3))))
Formula
a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k) * Catalan(n-3*k).
G.f.: 2/(1 + sqrt(1 - 4*x/(1 - x^3))).
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(-2*n+7)*a(n-3) +4*(n-5)*a(n-4) +(n-8)*a(n-6)=0. - R. J. Mathar, Oct 24 2024