A386837 - OEIS

%I #8 Aug 05 2025 10:03:51

%S 1,9,126,1978,32703,556887,9665476,170006256,3019802253,54047520709,

%T 973141183002,17607177876438,319855973830251,5830329608105763,

%U 106583422441886592,1953315343946213804,35875864591309216089,660185366847433991025,12169379986275311820790

%N a(n) = Sum_{k=0..n} binomial(4*n+2,k) * binomial(4*n-k-1,n-k).

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

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

%F a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+2,k) * binomial(n-k+2,n-k).

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

%o (PARI) a(n) = sum(k=0, n, binomial(4*n+2, k)*binomial(4*n-k-1, n-k));

%Y Cf. A386835, A386836.

%Y Cf. A370101, A386834.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Aug 05 2025