A382494 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(2*k,2*n-4*k).
1, 0, 3, 3, 6, 36, 16, 150, 165, 430, 1071, 1365, 4453, 6258, 14841, 29169, 49941, 115356, 190091, 404811, 750792, 1393956, 2808438, 4988268, 9905746, 18207126, 34231566, 65278964, 119255889, 227648406, 418394087, 782045001, 1457704212, 2681909302
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1500
- Index entries for linear recurrences with constant coefficients, signature (0,6,6,-15,-18,5,12,-3,32,12,-6,-4,18,-33,26,-15,6,-1).
Programs
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Magma
[&+[Binomial(k+2, 2)*Binomial(2*k, 2*n-4*k): k in [0..n]]: n in [0..41]]; // Vincenzo Librandi, May 11 2025
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Mathematica
Table[Sum[Binomial[k+2,2]*Binomial[2*k, 2*n-4*k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, May 11 2025 *) -
PARI
a(n) = sum(k=0, n\2, binomial(k+2, 2)*binomial(2*k, 2*n-4*k));
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PARI
my(N=2, M=40, x='x+O('x^M), X=1-x^2-x^3, Y=5); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))
Formula
G.f.: (Sum_{k=0..1} 4^k * binomial(3,2*k) * (1-x^2-x^3)^(3-2*k) * x^(5*k)) / ((1-x^2-x^3)^2 - 4*x^5)^3.
a(n) = 6*a(n-2) + 6*a(n-3) - 15*a(n-4) - 18*a(n-5) + 5*a(n-6) + 12*a(n-7) - 3*a(n-8) + 32*a(n-9) + 12*a(n-10) - 6*a(n-11) - 4*a(n-12) + 18*a(n-13) - 33*a(n-14) + 26*a(n-15) - 15*a(n-16) + 6*a(n-17) - a(n-18).