OFFSET
0,3
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.
FORMULA
a(n) = Sum_{k=0..n} 4^(n - k)*(n!/k!)*binomial(n-1, k-1).
Recurrence: a(n) = (8*n - 7)*a(n-1) - 16*(n-2)*(n-1)*a(n-2).
a(n) ~ n! * exp(sqrt(n) - 1/8) * 2^(2*n - 3/2) / (sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018
MAPLE
seq(coeff(series(factorial(n)*exp(x/(1-4*x)), x, n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018
MATHEMATICA
a[n_] := Sum[4^(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] = (8n - 7)*a[n - 1] - 16(n - 2)(n - 1) *a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)
CoefficientList[Series[Exp[x/(1 - 4*x)], {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Dec 07 2018 *)
PROG
(PARI) my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-4*x)))) \\ Michel Marcus, Nov 25 2018
(Maxima) (a[0] : 1, a[1] : 1, a[n] := (8*n - 7)*a[n-1] - 16*(n-2)*(n-1)*a[n-2], makelist(a[n], n, 0, 20)); /* Franck Maminirina Ramaharo, Nov 27 2018 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Ludovic Schwob, Nov 19 2018
STATUS
approved