A388203 - OEIS

%I #19 Sep 21 2025 23:54:13

%S 1,13,121,1000,7834,59719,448417,3337492,24708190,182319418,

%T 1342582762,9874401376,72571613716,533160730795,3916383099889,

%U 28768408020316,211347150876598,1552952656383862,11413589936099950,83907782844335248,617030596863871276

%N a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * binomial(n+3,k).

%H Vincenzo Librandi, <a href="/A388203/b388203.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = [x^n] (1-2*x)^n/(1-3*x)^(n+4).

%F a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n,k) * binomial(n+k+3,k).

%F a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(n+k+3,n).

%F G.f.: 1/(sqrt(1-8*x+4*x^2) * ((1-2*x + sqrt(1-8*x+4*x^2))/2)^3).

%F D-finite with recurrence n*(n+3)*a(n) +2*(-3*n^2-10*n-13)*a(n-1) +4*(-3*n^2+4*n-6)*a(n-2) +8*(n+1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Sep 16 2025

%F a(n) = [x^n] (1+x)^(n+3) * (3+x)^n. - _Seiichi Manyama_, Sep 21 2025

%t Table[Sum[ 3^k*Binomial[ n,k]*Binomial[n+3,k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 21 2025 *)

%o (PARI) a(n) = sum(k=0, n, 3^k*binomial(n, k)*binomial(n+3, k));

%o (Magma) [&+[3^k*Binomial(n, k)*Binomial(n+3,k): k in [0..n]]: n in [0..20]]; // _Vincenzo Librandi_, Sep 21 2025

%Y Cf. A069835, A388047, A388202, A388204.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Sep 15 2025