A361573 Expansion of e.g.f. exp(x^3/(6 * (1 - x)^3)).
1, 0, 0, 1, 12, 120, 1210, 13020, 152880, 1975960, 28148400, 440470800, 7525441000, 139375236000, 2778421245600, 59239029249400, 1343609515248000, 32274288638592000, 818014942318974400, 21809788600885084800, 610079100418595808000, 17863467401461938256000
Offset: 0
Keywords
Programs
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Mathematica
With[{nn=30},CoefficientList[Series[Exp[x^3/(6(1-x)^3)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 20 2023 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/(6*(1-x)^3)))) -
PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/6*sum(j=3, i, (-1)^(j-3)*j*binomial(-3, j-3)*v[i-j+1]/(i-j)!)); v;
Formula
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-1,n-3*k)/(6^k * k!).
a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=3..n} (-1)^(k-3) * k * binomial(-3,k-3) * a(n-k)/(n-k)!.
a(n) ~ 2^(-9/8) * exp(-1/24 + 5*2^(1/4)*n^(1/4)/48 - sqrt(2*n)/4 + 2^(7/4)*n^(3/4)/3 - n) * n^(n - 1/8) * (1 - 1203*2^(3/4)/(10240*n^(1/4))). - Vaclav Kotesovec, Mar 29 2023