A360152 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-5*k,n-3*k).
1, 2, 6, 21, 73, 262, 960, 3562, 13347, 50393, 191406, 730555, 2799622, 10765092, 41513751, 160490906, 621805286, 2413738744, 9385635299, 36550685683, 142534105563, 556514122937, 2175296066129, 8511430278018, 33334299581686, 130662787246407
Offset: 0
Keywords
Programs
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Maple
A360152 := proc(n) add(binomial(2*n-5*k,n-3*k),k=0..n/3) ; end proc: seq(A360152(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
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Mathematica
a[n_] := Sum[Binomial[2*n - 5*k, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *) -
PARI
a(n) = sum(k=0, n\3, binomial(2*n-5*k, n-3*k));
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PARI
my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-2*x^3/(1+sqrt(1-4*x)))))
Formula
G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3 * c(x)) ), where c(x) is the g.f. of A000108.
a(n) ~ 2^(2*n+5) / (31 * sqrt(Pi*n)). - Vaclav Kotesovec, Jan 28 2023
D-finite with recurrence 2*n*a(n) +4*(-2*n+1)*a(n-1) +(-3*n+4)*a(n-2) +2*(6*n-11)*a(n-3) +(n-4)*a(n-4) +2*(-n+9)*a(n-5) +4*(-2*n+1)*a(n-6) +(-n+4)*a(n-7) +2*(2*n-9)*a(n-8)=0. - R. J. Mathar, Mar 12 2023