A375565 a(n) = Sum_{k=0..floor(n/3)} (n-2*k+1) * binomial(n-2*k,k)^2.
1, 2, 3, 6, 17, 42, 90, 194, 441, 1006, 2242, 4950, 10974, 24376, 53961, 119048, 262337, 577782, 1271117, 2792718, 6129342, 13441616, 29454517, 64492800, 141108878, 308542280, 674238780, 1472532300, 3214268735, 7012637490, 15292425923, 33333338466, 72627184389
Offset: 0
Keywords
Programs
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Maple
A375565 := proc(n) add((n-2*k+1)*binomial(n-2*k,k)^2,k=0..floor(n/3)) ; end proc: seq(A375565(n),n=0..80) ; # R. J. Mathar, Oct 17 2024
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PARI
a(n) = sum(k=0, n\3, (n-2*k+1)*binomial(n-2*k, k)^2);
Formula
G.f.: (1-x-x^3)/((1-x-x^3)^2 - 4*x^4)^(3/2).
D-finite with recurrence 3*n*(n-1)*a(n) -(8*n-3)*(n-1)*a(n-1) +(7*n^2-14*n+8)*a(n-2) +(-8*n^2+3*n+23)*a(n-3) -2*n*(n+8)*a(n-4) +4*((n-1)^2)*a(n-5) +3*n*(n+2)*a(n-6) -2*n*(n-1)*a(n-7)=0. - R. J. Mathar, Oct 17 2024