A000772 E.g.f. exp(tan(x) + sec(x) - 1).

1, 1, 2, 6, 23, 107, 583, 3633, 25444, 197620, 1684295, 15618141, 156453857, 1683050189, 19344093070, 236497985706, 3063827565763, 41916787157011, 603799270943519, 9132945141812301, 144708157060239704, 2396568154933265024, 41403636316192616995

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

The number of elevated increasing binary trees. There is no restriction on the outdegree at the root. - Wenjin Woan, Jan 09 2008

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..478 (first 101 terms from T. D. Noe)
• Letong Hong and Rupert Li, Length-Four Pattern Avoidance in Inversion Sequences, arXiv:2112.15081 [math.CO], 2021.

Crossrefs

Cf. A000110, A094198, A147315, A185422.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(Tan(x) + Sec(x) - 1))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Jan 30 2020

Maple

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1,j-1)*b(j, 0), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, May 19 2021

Mathematica

nn = 25; Range[0, nn]! CoefficientList[Series[Exp[Tan[x] + Sec[x] - 1], {x, 0, nn}], x] (* T. D. Noe, Jun 20 2012 *)

Formula

a(n) = Sum_{k=1..n} A147315(n-1,k-1), n>0, a(0)=1. - Vladimir Kruchinin, Mar 10 2011

a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+x+x^2/2!)*d/dx. Cf. A000110 and A094198. See also A185422. - Peter Bala, Nov 25 2011

a(n) ~ 2^n * exp(2/Pi - 1 + 4*sqrt(n/Pi) - n) * n^(n - 1/4) / Pi^(n + 1/4). - Vaclav Kotesovec, Jan 27 2020