OFFSET
0,3
FORMULA
a(n) = n! * Sum_{k=0..n} (-2)^k * binomial(-(n-k)/2,k)/(n-k)! = n! * Sum_{k=0..n} 2^(n-k) * binomial(n-k/2-1,n-k)/k!.
From Vaclav Kotesovec, Feb 20 2024: (Start)
a(n) ~ 3^(-1/2) * 2^(n - 1/6) * exp(3*2^(-4/3)*n^(1/3) - n) * n^(n - 1/3) * (1 - 3/(16*(n/2)^(1/3))).
Recurrence (for n>5): (n-5)*a(n) = 3*(2*n^2 - 13*n + 16)*a(n-1) - (12*n^3 - 108*n^2 + 299*n - 259)*a(n-2) + (n-2)*(8*n^3 - 84*n^2 + 290*n - 327)*a(n-3) + (n-4)*(n-3)*(n-2)*a(n-4). (End)
MATHEMATICA
Table[n! * Sum[2^(n-k) * Binomial[n-k/2-1, n-k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 20 2024 *)
Join[{1, 1}, RecurrenceTable[{(-4 + n) (-3 + n) (-2 + n) a[-4 + n] + (-2 + n) (-327 + 290 n - 84 n^2 + 8 n^3) a[-3 + n] + (259 - 299 n + 108 n^2 - 12 n^3) a[-2 + n] + 3 (16 - 13 n + 2 n^2) a[-1 + n] + (5 - n) a[n] == 0, a[2] == 3, a[3] == 16, a[4] == 121, a[5] == 1176}, a, {n, 2, 20}]] (* Vaclav Kotesovec, Feb 20 2024 *)
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/sqrt(1-2*x))))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Apr 11 2023
STATUS
approved