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A382494 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(2*k,2*n-4*k).

%I A382494 #19 May 11 2025 22:05:08
%S A382494 1,0,3,3,6,36,16,150,165,430,1071,1365,4453,6258,14841,29169,49941,
%T A382494 115356,190091,404811,750792,1393956,2808438,4988268,9905746,18207126,
%U A382494 34231566,65278964,119255889,227648406,418394087,782045001,1457704212,2681909302
%N A382494 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(2*k,2*n-4*k).
%H A382494 Vincenzo Librandi, <a href="/A382494/b382494.txt">Table of n, a(n) for n = 0..1500</a>
%H A382494 <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,6,-15,-18,5,12,-3,32,12,-6,-4,18,-33,26,-15,6,-1).
%F A382494 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.
%F A382494 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).
%t A382494 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 *)
%o A382494 (PARI) a(n) = sum(k=0, n\2, binomial(k+2, 2)*binomial(2*k, 2*n-4*k));
%o A382494 (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))
%o A382494 (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
%Y A382494 Cf. A376729, A382300, A382495.
%Y A382494 Cf. A034839, A377146, A382230.
%K A382494 nonn,easy
%O A382494 0,3
%A A382494 _Seiichi Manyama_, Mar 29 2025