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A381421 a(n) = Sum_{k=0..n} (k+1) * binomial(2*k,2*n-2*k).

%I A381421 #44 Apr 23 2025 10:47:06
%S A381421 1,2,5,22,68,206,631,1870,5467,15836,45416,129260,365565,1028122,
%T A381421 2877697,8021010,22274476,61653850,170152275,468347046,1286055927,
%U A381421 3523777912,9635982160,26302324504,71674754873,195015074610,529846108989,1437657038030,3896050721940
%N A381421 a(n) = Sum_{k=0..n} (k+1) * binomial(2*k,2*n-2*k).
%H A381421 Vincenzo Librandi, <a href="/A381421/b381421.txt">Table of n, a(n) for n = 0..500</a>
%H A381421 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,0,-11,0,-2,4,-1).
%F A381421 G.f.: ((1-x-x^2)^2 + 4*x^3) / ((1-x-x^2)^2 - 4*x^3)^2.
%F A381421 a(n) = 4*a(n-1) - 2*a(n-2) - 11*a(n-4) - 2*a(n-6) + 4*a(n-7) - a(n-8).
%t A381421 Table[Sum[(k+1)*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Apr 23 2025 *)
%o A381421 (PARI) a(n) = sum(k=0, n, (k+1)*binomial(2*k, 2*n-2*k));
%o A381421 (PARI) my(N=1, 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 A381421 (Magma) [&+[(k+1) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // _Vincenzo Librandi_, Apr 23 2025
%Y A381421 Cf. A108479, A382230, A382470, A382471, A382472, A382473, A382474.
%Y A381421 Cf. A034839.
%K A381421 nonn,easy
%O A381421 0,2
%A A381421 _Seiichi Manyama_, Mar 28 2025