A388206 - OEIS

%I #19 Sep 21 2025 23:54:20

%S 1,17,201,2073,20049,187425,1718425,15572009,140075937,1254065201,

%T 11192366953,99684240441,886620991473,7878859270209,69975461449017,

%U 621277911345225,5515122857993025,48955858259392593,434581254318819849,3858161907813449817,34257197676589542801

%N a(n) = Sum_{k=0..n} 4^k * binomial(n,k) * binomial(n+3,k).

%H Vincenzo Librandi, <a href="/A388206/b388206.txt">Table of n, a(n) for n = 0..500</a>

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

%F a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(n,k) * binomial(n+k+3,k).

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

%F G.f.: 1/(sqrt(1-10*x+9*x^2) * ((1-3*x + sqrt(1-10*x+9*x^2))/2)^3).

%F D-finite with recurrence n*(n+3)*a(n) +(-7*n^2-25*n-36)*a(n-1) +3*(-7*n^2+11*n-18)*a(n-2) +27*(n+1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Sep 16 2025

%F a(n) = [x^n] (1+x)^(n+3) * (4+x)^n. - _Seiichi Manyama_, Sep 21 2025

%t Table[Sum[4^k*Binomial[n,k]*Binomial[n+3,k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Sep 17 2025 *)

%o (PARI) a(n) = sum(k=0, n, 4^k*binomial(n, k)*binomial(n+3, k));

%o (Magma) [&+[4^k*Binomial(n, k)*Binomial(n+3,k): k in [0..n]]: n in [0..20]]; // _Vincenzo Librandi_, Sep 17 2025

%Y Cf. A084771, A388060, A388205, A388207.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Sep 15 2025