A373770 - OEIS

%I #10 Jun 18 2024 10:01:25

%S 1,1,3,12,63,405,3075,26880,265545,2922885,35447895,469396620,

%T 6736095135,104102463465,1723322736135,30416726340000,570089983287825,

%U 11306156398562025,236514323713142475,5204122351983254700,120139520273298100575,2903216115946088267325

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

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

%F From _Vaclav Kotesovec_, Jun 18 2024: (Start)

%F Recurrence: 2*a(n) = 2*(2*n-1)*a(n-1) - 2*(n-2)*(n-1)*a(n-2) - (n-2)*(n-1)*a(n-3).

%F a(n) ~ 2^(-1/4) * exp(-3/4 + sqrt(2*n) - n) * n^(n + 1/4) * (1 + 7/(6*sqrt(2*n))). (End)

%o (PARI) a(n) = n!*sum(k=0, n\2, binomial(n-k, n-2*k)/(2^k*k!));

%Y Cf. A130905, A361596, A373771.

%Y Cf. A185369.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jun 18 2024