OFFSET
0,4
COMMENTS
a(n) = Sum_{k>=0} k*A186368(n,k).
LINKS
E. Deutsch and S. Elizalde, Cycle up-down permutations, arXiv:0909.5199 [math.CO], 2009.
FORMULA
E.g.f.: [z(1+sin z)-cos z * log(sec z + tan z)]/[2 cos z *(1-sin z)].
a(n) ~ n! * n^2 * (2/Pi)^(n+2) * (1 - log(n)/n). - Vaclav Kotesovec, Aug 23 2014
EXAMPLE
a(3) = 4 because the permutations (1)(2)(3), (1)(23), (12)(3), (13)(2), (132) have a total of 0+1+1+1+1=4 excedances.
MAPLE
g := ((z*(1+sin(z))-cos(z)*ln(sec(z)+tan(z)))*1/2)/(cos(z)*(1-sin(z))): gser := series(g, z = 0, 30): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 22);
MATHEMATICA
T[n_, k_] := n! SeriesCoefficient[(Sec[z Sqrt[t]]+ Tan[z Sqrt[t]])^( 1/Sqrt[t])/Cos[z Sqrt[t]], {z, 0, n}, {t, 0, k}];
Table[Sum[k T[n, k], {k, 0, n/2}], {n, 0, 22}] (* Jean-François Alcover, Aug 07 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 28 2011
STATUS
approved