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A386955 a(n) = Sum_{k=0..n} (k+1) * 2^k * binomial(2*n+1,n-k).

%I A386955 #17 Aug 13 2025 15:00:17
%S A386955 1,7,42,235,1262,6594,33780,170475,850230,4200130,20585228,100220718,
%T A386955 485164988,2337145360,11210274408,53567616267,255110184486,
%U A386955 1211287208346,5735765695260,27093982041546,127699233939684,600650635811532,2819989050992472,13216897613555550
%N A386955 a(n) = Sum_{k=0..n} (k+1) * 2^k * binomial(2*n+1,n-k).
%H A386955 Vincenzo Librandi, <a href="/A386955/b386955.txt">Table of n, a(n) for n = 0..400</a>
%F A386955 a(n) = [x^n] 1/((1-3*x)^2 * (1-x)^n).
%F A386955 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k-1,n-k).
%F A386955 a(n) = Sum_{k=0..n} (k+1) * 3^k * binomial(2*n-k-1,n-k).
%F A386955 G.f.: 2 * (1+sqrt(1-4*x))/( sqrt(1-4*x) * (3*sqrt(1-4*x)-1)^2 ).
%F A386955 a(n) ~ n * 3^(2*n) / 2^(n+1). - _Vaclav Kotesovec_, Aug 12 2025
%t A386955 Table[Sum[(k+1)*2^k*Binomial[2*n+1,n-k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 12 2025 *)
%o A386955 (PARI) a(n) = sum(k=0, n, (k+1)*2^k*binomial(2*n+1, n-k));
%o A386955 (Magma) [&+[(k+1) * 2^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 12 2025
%Y A386955 Cf. A088218, A258431, A384365, A386956.
%K A386955 nonn
%O A386955 0,2
%A A386955 _Seiichi Manyama_, Aug 11 2025