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A387337 a(n) = Sum_{k=0..n} 2^k * binomial(n+2,k) * binomial(n+2,k+2).

%I A387337 #18 Aug 29 2025 20:55:41
%S A387337 1,9,62,390,2355,13923,81340,471852,2725365,15698925,90278298,
%T A387337 518620050,2977523367,17089401735,98073111800,562837648728,
%U A387337 3230477449641,18545060127249,106484493098230,611580138259230,3513476242850651,20190255588746859,116056618927038132
%N A387337 a(n) = Sum_{k=0..n} 2^k * binomial(n+2,k) * binomial(n+2,k+2).
%H A387337 Vincenzo Librandi, <a href="/A387337/b387337.txt">Table of n, a(n) for n = 0..1000</a>
%F A387337 n*(n+4)*a(n) = (n+2) * (3*(2*n+3)*a(n-1) - (n+1)*a(n-2)) for n > 1.
%F A387337 a(n) = Sum_{k=0..floor(n/2)} 2^k * 3^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
%F A387337 a(n) = [x^n] (1+3*x+2*x^2)^(n+2).
%F A387337 E.g.f.: exp(3*x) * BesselI(2, 2*sqrt(2)*x) / 2, with offset 2.
%F A387337 a(n) ~ (1 + sqrt(2))^(2*n+5) / (2^(9/4) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 27 2025
%t A387337 Table[Sum[2^k*Binomial[n+2, k] * Binomial[n+2, k+2], {k,0,n}], {n,0,25}] (* _Vaclav Kotesovec_, Aug 27 2025 *)
%o A387337 (PARI) a(n) = sum(k=0, n, 2^k*binomial(n+2, k)*binomial(n+2, k+2));
%o A387337 (Magma) [&+[2^k * Binomial(n+2,k) * Binomial(n+2,k+2): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 29 2025
%Y A387337 Cf. A050151.
%K A387337 nonn,new
%O A387337 0,2
%A A387337 _Seiichi Manyama_, Aug 27 2025