A361598 - OEIS

%I #11 Mar 17 2023 08:30:30

%S 1,2,9,58,473,4626,52537,677594,9762993,155175778,2693718281,

%T 50657791482,1025158123849,22198908725618,511885585833273,

%U 12517101011344666,323402336324055137,8800318580852865474,251497162228635927433,7529081846683064675258

%N Expansion of e.g.f. exp( x/(1-x)^2 ) / (1-x).

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

%F From _Vaclav Kotesovec_, Mar 17 2023: (Start)

%F a(n) = (3*n - 1)*a(n-1) - (n-1)*(3*n - 5)*a(n-2) + (n-2)^2*(n-1)*a(n-3).

%F a(n) ~ 2^(-1/6) * 3^(-1/2) * exp(-1/12 + 3*2^(-2/3)*n^(2/3) - n) * n^(n + 1/6) * (1 + 1/(2^(2/3)*n^(1/3)) + 83/(360*2^(1/3)*n^(2/3))). (End)

%t Table[n! * Sum[Binomial[n+k,2*k]/k!, {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 17 2023 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^2)/(1-x)))

%Y Column k=2 of A361600.

%Y Cf. A082579.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 17 2023