A395216 - OEIS

%I #13 Apr 30 2026 04:45:25

%S 1,9,681,148905,64805265,46896669225,50831084252025,77061447551313225,

%T 155672917524626126625,404153655359180645543625,

%U 1311161072449806104077175625,5197766162779913140792304555625,24721993179513170562351592437800625,138944861938899421852917299018079515625

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

%F Let A(n,k) = (2*n)! * [x^(2*n)] 1/((1 - x)*(1 - 3*x))^(k/2). A(0,k) = 1 and A(n,k) = k*(k+2) * A(n-1,k+4) + 3*k*(k+1) * A(n-1,k+2) for n > 0. a(n) = A(n,1).

%F a(n) = (2*n)! * Sum_{k=0..n} 16^k * (-3)^(n-k) * binomial(n+k-1/2,n+k) * binomial(n+k,2*k).

%F a(n) ~ 2^(2*n) * 3^(2*n + 1/2) * n^(2*n) / exp(2*n). - _Vaclav Kotesovec_, Apr 30 2026

%t Table[(-3)^n * Binomial[n - 1/2, n] * (2*n)! * Hypergeometric2F1[-n, 1/2 + n, 1/2, 4/3], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 30 2026 *)

%o (PARI) a(n) = my(x='x+O('x^(2*n+1))); (2*n)!*polcoef(1/sqrt((1-x)*(1-3*x)), 2*n);

%Y Column k=1 of A395215.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 16 2026