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A382496 a(n) = Sum_{k=0..floor(n/3)} (k+1) * binomial(2*k,2*n-6*k).

%I A382496 #19 May 12 2025 10:13:53
%S A382496 1,0,0,2,2,0,3,18,3,4,60,60,9,140,350,146,275,1260,1267,732,3471,6476,
%T A382496 4193,8470,24040,25104,24388,72810,117368,102672,202031,440750,490884,
%U A382496 612012,1419042,2121626,2281049,4267188,7951185,9511604,13402924,26600984,38465043,47376620
%N A382496 a(n) = Sum_{k=0..floor(n/3)} (k+1) * binomial(2*k,2*n-6*k).
%H A382496 Vincenzo Librandi, <a href="/A382496/b382496.txt">Table of n, a(n) for n = 0..1500</a>
%H A382496 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,4,4,0,-6,-4,-6,4,-4,-4,3,4,-6,4,-1).
%F A382496 G.f.: ((1-x^3-x^4)^2 + 4*x^7) / ((1-x^3-x^4)^2 - 4*x^7)^2.
%F A382496 a(n) = 4*a(n-3) + 4*a(n-4) - 6*a(n-6) - 4*a(n-7) - 6*a(n-8) + 4*a(n-9) - 4*a(n-10) - 4*a(n-11) + 3*a(n-12) + 4*a(n-13) - 6*a(n-14) + 4*a(n-15) - a(n-16).
%t A382496 Table[Sum[(k+1)*Binomial[2*k, 2*n-6*k],{k,0,Floor[n/3]}],{n,0,43}] (* _Vincenzo Librandi_, May 12 2025 *)
%o A382496 (PARI) a(n) = sum(k=0, n\3, (k+1)*binomial(2*k, 2*n-6*k));
%o A382496 (PARI) my(N=1, M=50, x='x+O('x^M), X=1-x^3-x^4, Y=7); 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 A382496 (Magma) [&+[(k+1)*Binomial(2*k, 2*n-6*k): k in [0..n]]: n in [0..45]]; // _Vincenzo Librandi_, May 12 2025
%Y A382496 Cf. A381421, A382300.
%Y A382496 Cf. A034839, A375470.
%K A382496 nonn,easy
%O A382496 0,4
%A A382496 _Seiichi Manyama_, Mar 29 2025