A321837 Expansion of e.g.f.: exp(x/(1-3*x)).

1, 1, 7, 73, 1009, 17341, 355951, 8488117, 230439553, 7013527129, 236419161751, 8740611892321, 351566026652017, 15280473017519893, 713558666964639679, 35623071889296787981, 1893073661362838712961, 106682309871314293118257

Offset: 0

Author

Ludovic Schwob, Nov 19 2018

Keywords

• nonn

Comments

For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.

Links

• Ludovic Schwob, Table of n, a(n) for n = 0..199
• Norihiro Nakashima, Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Crossrefs

Cf. A000262, A025168, A321847, A321848, A321849, A321850 (analogs for k=1,2,4,5,6,7).

Programs

GAP

Concatenation([1], List([1..25], n-> Sum([1..n], k-> 3^(n-k)*(Factorial(n)/Factorial(k))*Binomial(n-1, k-1)))); # G. C. Greubel, Dec 14 2018

Magma

[1] cat [&+[3^(n-k)*Factorial(n) div Factorial(k)*Binomial(n-1, k-1):  k in [0..n]]: n in [1.. 18]]; // Vincenzo Librandi, Dec 08 2018

Maple

seq(coeff(series(factorial(n)*exp(x/(1-3*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018

Mathematica

a[n_] := Sum[3^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (6n - 5)*a[n - 1] - 9(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)

PARI

my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-3*x)))) \\ Michel Marcus, Nov 25 2018

Sage

{c[1]:c[0]*factorial(c[1]) for c in (exp(x/(1-3*x))).taylor(x,0,25).coefficients()} # G. C. Greubel, Dec 14 2018

Formula

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

Recurrence: a(n) = (6*n-5)*a(n-1) - 9*(n-2)*(n-1)*a(n-2).

a(n) ~ n! * exp(2*sqrt(n/3) - 1/6) * 3^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018