OFFSET
0,3
LINKS
Masato Kobayashi, A new refinement of Euler numbers on counting alternating permutations, arXiv:1908.00701 [math.CO], 2019.
FORMULA
a(n-2) = |{up-down 2nd-max-upper permutations in S_n}| for n >= 2 (see Definition 3.4 in Kobayashi).
a(0) = 0 and a(n) = 2*A000142(n)*Sum_{i,j,k>=0, (2*i+1)+(2*j+1)+k=n} A000111(2*i+1)*A000111(2*j+1)*A000111(k)/(A000142(2*i+1)*A000142(2*j+1)*A000142(k)) for n > 0 (see Lemma 3.6 in Kobayashi).
a(2*n) = 2*A225689(2*n) (see Lemma 4.2 in Kobayashi).
a(n) ~ n! * 2^(n+4) * n^2 / Pi^(n+3). - Vaclav Kotesovec, Aug 12 2019
MAPLE
gf := (2*sin(x)*tan(x))/(1 - sin(x)): ser := series(gf, x, 25):
seq(n!*coeff(ser, x, n), n=0..23); # Peter Luschny, Aug 19 2019
MATHEMATICA
CoefficientList[Series[2Tan[x]^2(Sec[x]+Tan[x]), {x, 0, 23}], x]*Table[n!, {n, 0, 23}]
PROG
(PARI) my(x='x+O('x^30)); concat([0, 0], Vec(serlaplace(2*(tan(x))^2*(1/cos(x) + tan(x))))) \\ Michel Marcus, Aug 13 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Stefano Spezia, Aug 12 2019
STATUS
approved