A321848 E.g.f.: exp(x/(1-5*x)).

1, 1, 11, 181, 3961, 108101, 3532651, 134415961, 5834249681, 284391878761, 15378011541451, 913297438474301, 59086483931657161, 4135583008765323181, 311324086814794408811, 25079793551003791168801, 2152597370423901820231201, 196089415332225446044417361

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, A321837, A321847, A321849, A321850.

Programs

Maple

seq(coeff(series(factorial(n)*exp(x/(1-5*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018

Mathematica

a[n_] := Sum[5^(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] = (10 n - 9)*a[n - 1] - 25(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)
With[{nn=20},CoefficientList[Series[Exp[x/(1-5x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 19 2024 *)

PARI

my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-5*x)))) \\ Michel Marcus, Nov 25 2018

Formula

a(n) = Sum_{k=0..n} 5^(n-k)*(n!/k!)*binomial(n-1, k-1).

Recurrence: a(n) = (10*n-9)*a(n-1) - 25*(n-2)*(n-1)*a(n-2).

a(n) ~ n! * exp(2*sqrt(n/5) - 1/10) * 5^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018