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A382230 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(2*k,2*n-2*k).

%I A382230 #32 Apr 22 2025 11:05:40
%S A382230 1,3,9,46,171,591,2033,6714,21606,68308,212370,651234,1974113,5924277,
%T A382230 17623671,52025858,152539077,444530073,1288396257,3715833732,
%U A382230 10668907932,30507914696,86912853588,246755125332,698353551105,1970673504951,5545952371509,15568330002486
%N A382230 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(2*k,2*n-2*k).
%H A382230 Vincenzo Librandi, <a href="/A382230/b382230.txt">Table of n, a(n) for n = 0..500</a>
%H A382230 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,2,-18,30,7,30,-18,2,-9,6,-1).
%F A382230 G.f.: (Sum_{k=0..1} 4^k * binomial(3,2*k) * (1-x-x^2)^(3-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^3.
%F A382230 a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3) - 18*a(n-4) + 30*a(n-5) + 7*a(n-6) + 30*a(n-7) - 18*a(n-8) + 2*a(n-9) - 9*a(n-10) + 6*a(n-11) - a(n-12).
%t A382230 Table[Sum[Binomial[k+2,2]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Apr 22 2025 *)
%o A382230 (PARI) a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(2*k, 2*n-2*k));
%o A382230 (PARI) my(N=2, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); 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 A382230 (Magma) [&+[Binomial(k+2, 2) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // _Vincenzo Librandi_, Apr 22 2025
%Y A382230 Cf. A108479, A381421, A382470, A382471, A382472, A382473, A382474.
%Y A382230 Cf. A034839, A377145.
%K A382230 nonn,easy
%O A382230 0,2
%A A382230 _Seiichi Manyama_, Mar 28 2025