A361545 Expansion of e.g.f. exp(x^4/(24 * (1-x))).
1, 0, 0, 0, 1, 5, 30, 210, 1715, 15750, 160650, 1801800, 22043175, 292116825, 4168464300, 63725161500, 1039028615625, 17998106626500, 330068683444500, 6388785205803000, 130156170633113625, 2783924007745505625, 62375052003905891250, 1460924768552182683750
Offset: 0
Programs
-
Mathematica
RecurrenceTable[{3 (-4 + n) (-3 + n) (-2 + n) (-1 + n) a[-5 + n] - 4 (-3 + n) (-2 + n) (-1 + n) a[-4 + n] + 24 (-2 + n) (-1 + n) a[-2 + n] - 48 (-1 + n) a[-1 + n] + 24 a[n] == 0, a[1] == 0, a[2] == 0, a[3] == 0, a[4] == 1, a[5] == 5}, a, {n, 0, 25}] (* Vaclav Kotesovec, Aug 28 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4/(24*(1-x)))))
Formula
a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + binomial(n-1,3) * a(n-4) - 3*binomial(n-1,4) * a(n-5) for n > 4.
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k)/(24^k * k!).
a(0) = 1; a(n) = ((n-1)!/24) * Sum_{k=4..n} k * a(n-k)/(n-k)!. (End)
a(n) ~ 2^(-5/4) * 3^(-1/4) * exp(-7/48 + sqrt(n/6) - n) * n^(n - 1/4). - Vaclav Kotesovec, Aug 28 2025