A361572 Expansion of e.g.f. exp( (x / (1-x))^3 ).

1, 0, 0, 6, 72, 720, 7560, 90720, 1270080, 20381760, 364694400, 7125148800, 150186960000, 3393726336000, 81882210009600, 2102315389574400, 57244753133568000, 1647544166940672000, 49957730917981286400, 1591303422125646028800

Offset

0,4

Links

Table of n, a(n) for n=0..19.

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-1,n-3*k)/k!.

a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} (-1)^(k-3) * k * binomial(-3,k-3) * a(n-k)/(n-k)!.

From Vaclav Kotesovec, Mar 17 2023: (Start)

a(n) = 4*(n-1)*a(n-1) - 6*(n-2)*(n-1)*a(n-2) + (n-2)*(n-1)*(4*n - 9)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4).

a(n) ~ 3^(1/8) * exp(-1/4 + 5*3^(-1/4)*n^(1/4)/8 - sqrt(3*n)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n - 1/8) / 2 * (1 - (409/2560)*3^(1/4)/n^(1/4)). (End)

Mathematica

Table[n! * Sum[Binomial[n-1, n-3*k]/k!, {k, 0, n/3}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 17 2023 *)
With[{nn=20}, CoefficientList[Series[Exp[(x/(1-x))^3], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Sep 07 2024 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((x/(1-x))^3)))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, (-1)^(j-3)*j*binomial(-3, j-3)*v[i-j+1]/(i-j)!)); v;

Crossrefs

Cf. A000262, A052887, A361576.

Cf. A091695.

Sequence in context: A059410 A238103 A200978 * A099673 A105928 A186665

Adjacent sequences: A361569 A361570 A361571 * A361573 A361574 A361575

Keyword

nonn

Author

Seiichi Manyama, Mar 16 2023

Status

approved