cp's OEIS Frontend

A387309 a(n) = Sum_{k=0..n} 3^k * binomial(n+1,k+1) * binomial(2*k+2,k+2).

%I A387309 #19 Aug 30 2025 10:26:16
%S A387309 1,14,174,2128,26045,320082,3951493,48987848,609592347,7610525650,
%T A387309 95287524332,1196054790168,15046318739803,189654839753750,
%U A387309 2394743468261190,30285593026553536,383554551776056139,4863775493104574634,61748210178809072722,784757334938247965840,9983152795673915802399
%N A387309 a(n) = Sum_{k=0..n} 3^k * binomial(n+1,k+1) * binomial(2*k+2,k+2).
%H A387309 Vincenzo Librandi, <a href="/A387309/b387309.txt">Table of n, a(n) for n = 0..800</a>
%F A387309 G.f.: ((1-7*x)/sqrt((1-x) * (1-13*x)) - 1)/(18*x^2).
%F A387309 n*(n+2)*a(n) = (n+1) * (7*(2*n+1)*a(n-1) - 13*n*a(n-2)) for n > 1.
%F A387309 a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
%F A387309 a(n) = [x^n] (1+7*x+9*x^2)^(n+1).
%F A387309 E.g.f.: exp(7*x) * BesselI(1, 6*x) / 3, with offset 1.
%t A387309 Table[Sum[3^k*Binomial[n+1,k+1]*Binomial[2*k+2,k+2],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 30 2025 *)
%o A387309 (PARI) a(n) = sum(k=0, n, 3^k*binomial(n+1, k+1)*binomial(2*k+2, k+2));
%o A387309 (Magma) [&+[3^k * Binomial(n+1,k+1) * Binomial(2*k+2,k+2): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 30 2025
%Y A387309 Cf. A387310, A387311.
%Y A387309 Cf. A026376, A331793.
%Y A387309 Cf. A385716.
%K A387309 nonn,new
%O A387309 0,2
%A A387309 _Seiichi Manyama_, Aug 25 2025