A361597 Expansion of e.g.f. exp( x^3/(6 * (1-x)^3) ) / (1-x).
1, 1, 2, 7, 40, 320, 3130, 34930, 432320, 5866840, 86816800, 1395455600, 24270908200, 454897042600, 9146979842000, 196443726879400, 4486709145318400, 108548344109004800, 2771885136281060800, 74475606190225240000, 2099591224223100608000
Offset: 0
Keywords
Programs
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Mathematica
Table[n! * Sum[Binomial[n,3*k]/(6^k * k!), {k,0,n/3}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/(6*(1-x)^3))/(1-x)))
Formula
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n,3*k)/(6^k * k!).
From Vaclav Kotesovec, Mar 17 2023: (Start)
Recurrence: 2*a(n) = 2*(4*n - 3)*a(n-1) - 6*(n-1)*(2*n - 3)*a(n-2) + (n-2)*(n-1)*(8*n - 17)*a(n-3) - 2*(n-3)^2*(n-2)*(n-1)*a(n-4).
a(n) ~ 2^(-7/8) * exp(-1/24 + 5*2^(-15/4)*n^(1/4)/3 - sqrt(n/2)/2 + 2^(7/4)*n^(3/4)/3 - n) * n^(n + 1/8) * (1 + (2637/10240)*2^(3/4)/n^(1/4)). (End)