cp's OEIS Frontend

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

%I A000772 #41 Sep 08 2022 08:44:28
%S A000772 1,1,2,6,23,107,583,3633,25444,197620,1684295,15618141,156453857,
%T A000772 1683050189,19344093070,236497985706,3063827565763,41916787157011,
%U A000772 603799270943519,9132945141812301,144708157060239704,2396568154933265024,41403636316192616995
%N A000772 E.g.f. exp(tan(x) + sec(x) - 1).
%C A000772 The number of elevated increasing binary trees. There is no restriction on the outdegree at the root. - _Wenjin Woan_, Jan 09 2008
%H A000772 Alois P. Heinz, <a href="/A000772/b000772.txt">Table of n, a(n) for n = 0..478</a> (first 101 terms from T. D. Noe)
%H A000772 Letong Hong and Rupert Li, <a href="https://arxiv.org/abs/2112.15081">Length-Four Pattern Avoidance in Inversion Sequences</a>, arXiv:2112.15081 [math.CO], 2021.
%F A000772 a(n) = Sum_{k=1..n} A147315(n-1,k-1), n>0, a(0)=1. - _Vladimir Kruchinin_, Mar 10 2011
%F A000772 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
%F A000772 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
%p A000772 b:= proc(u, o) option remember;
%p A000772       `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
%p A000772     end:
%p A000772 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A000772       a(n-j)*binomial(n-1,j-1)*b(j, 0), j=1..n))
%p A000772     end:
%p A000772 seq(a(n), n=0..23);  # _Alois P. Heinz_, May 19 2021
%t A000772 nn = 25; Range[0, nn]! CoefficientList[Series[Exp[Tan[x] + Sec[x] - 1], {x, 0, nn}], x] (* _T. D. Noe_, Jun 20 2012 *)
%o A000772 (Magma) m:=25; R<x>:=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
%Y A000772 Cf. A000110, A094198, A147315, A185422.
%K A000772 nonn
%O A000772 0,3
%A A000772 _N. J. A. Sloane_