OFFSET
0,4
FORMULA
a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-3*k) * binomial(n-1-3*k/2,n-2*k)/k!.
From Vaclav Kotesovec, Feb 20 2024: (Start)
Recurrence (for n>9): 4*(n-9)*(n-2)*a(n) = 12*(n-1)*(2*n^2 - 23*n + 48)*a(n-1) - 48*(n-2)*(n-1)*(n^2 - 12*n + 30)*a(n-2) + 16*(n-8)*(n-3)*(n-2)*(n-1)*(2*n - 9)*a(n-3) + 4*(n-9)*(n-3)*(n-2)*(n-1)*a(n-4) - 6*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 17)*a(n-5) + 9*(n-8)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6).
a(n) ~ 2^(n - 5/6) * exp(3*2^(-8/3)*n^(1/3) - n) * n^(n - 1/3) / sqrt(3) * (1 - 7*2^(2/3)/(128*n^(1/3))). (End)
MATHEMATICA
Flatten[{{1, 0, 1, 3}, RecurrenceTable[{-9 (-8 + n) (-5 + n) (-4 + n) (-3 + n) (-2 + n) (-1 + n) a[-6 + n] + 6 (-4 + n) (-3 + n) (-2 + n) (-1 + n) (-17 + 2 n) a[-5 + n] - 4 (-9 + n) (-3 + n) (-2 + n) (-1 + n) a[-4 + n] - 16 (-8 + n) (-3 + n) (-2 + n) (-1 + n) (-9 + 2 n) a[-3 + n] + 48 (-2 + n) (-1 + n) (30 - 12 n + n^2) a[-2 + n] - 12 (-1 + n) (48 - 23 n + 2 n^2) a[-1 + n] + 4 (-9 + n) (-2 + n) a[n] == 0, a[4] == 21, a[5] == 180, a[6] == 1950, a[7] == 25200, a[8] == 378105, a[9] == 6452460}, a, {n, 4, 20}]}] (* Vaclav Kotesovec, Feb 19 2024 *)
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x^2/(2*sqrt(1-2*x)))))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Feb 06 2024
STATUS
approved