A309845 Expansion of e.g.f.: sec(x) + 2*tan(x).

1, 2, 1, 4, 5, 32, 61, 544, 1385, 15872, 50521, 707584, 2702765, 44736512, 199360981, 3807514624, 19391512145, 419730685952, 2404879675441, 58177770225664, 370371188237525, 9902996106248192, 69348874393137901, 2030847773013704704, 15514534163557086905, 493842960380415967232

Offset: 0

Author

Stefano Spezia, Aug 20 2019

Keywords

• nonn
• easy

Links

• F. Heneghan and T. K. Petersen, Power series for up-down min-max permutations, 2013.
• Masato Kobayashi, A new refinement of Euler numbers on counting alternating permutations, arXiv:1908.00701 [math.CO], 2019.

Crossrefs

Cf. A000111, A000182, A000364, A164575, A181937, A225688, A225689.

Programs

Mathematica

CoefficientList[Series[Sec[x]+2Tan[x],{x,0,25}],x]*Table[n!,{n,0,25}]

PARI

my(x='x+O('x^30)); Vec(serlaplace(1/cos(x)+2*tan(x))) \\ Michel Marcus, Aug 20 2019

Formula

a(n-2) = |{up-down 2nd-max-lower permutations in S_n}| for n >= 2 (see Definition 3.4 in Kobayashi).

a(n) = A000111(n+2) - A164575(n) (See Definition 3.4 in Kobayashi).

a(n) = A225688(n) + A225689(n) - A164575(n) (See Heneghan-Petersen and Kobayashi articles).

a(2*n) = A000111(2*n) (See Lemma 3.8 in Kobayashi).

a(2*n+1) = 2*A000111(2*n+1) (See Lemma 3.8 in Kobayashi).