A388129 - OEIS

%I #23 Nov 09 2025 09:38:47

%S 1,5,34,253,1966,15674,127100,1043165,8639350,72053110,604304636,

%T 5091508690,43061767516,365373061772,3108708313744,26513185631965,

%U 226595976712870,1940187521222222,16639766719758764,142918245750677158,1229139650429018852,10583595038040435020

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

%H Vincenzo Librandi, <a href="/A388129/b388129.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = [x^n] (1+x)^n/(1-x)^(2*n+2).

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

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

%F D-finite with recurrence 16*n*(2*n+1)*a(n) +4*(-81*n^2+25*n-4)*a(n-1) +(371*n^2-1073*n+782)*a(n-2) -10*(2*n-3)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Sep 16 2025

%F a(n) = [x^n] (1+x)^(2*n+1) * (1+2*x)^n. - _Seiichi Manyama_, Sep 21 2025

%F Recurrence (of order 2): 4*n*(2*n + 1)*(17*n - 11)*a(n) = (1207*n^3 - 781*n^2 - 162*n + 96)*a(n-1) - 2*(n-1)*(2*n - 1)*(17*n + 6)*a(n-2). - _Vaclav Kotesovec_, Nov 09 2025

%t Table[Sum[ 2^(n-k)* Binomial[ n,k]*Binomial[2*n+1,k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 24 2025 *)

%o (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(n, k)*binomial(2*n+1, k));

%o (Magma) [&+[2^(n-k)*Binomial(n, k)*Binomial(2*n+1, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 24 2025

%Y Cf. A388130, A388131.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Sep 14 2025