%I A040027
%S A040027 1,1,3,9,31,121,523,2469,12611,69161,404663,2512769,16485691,
%T A040027 113842301,824723643,6249805129,49416246911,406754704841,3478340425563,
%U A040027 30845565317189,283187362333331,2687568043654521,26329932233283223
%N A040027 Second-from-right diagonal of triangle A121207.
%C A040027 Number of permutations beginning with 21 and avoiding 1-23. - Ralf Stephan, 
               Apr 25 2004
%C A040027 Originally defined as main diagonal of an array of binomial recurrence 
               coefficients (see Gould).
%C A040027 Starting (1, 3, 9, 31, 121,...) = row sums of triangle A153868 [From 
               Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 03 2009]
%C A040027 Equals eigensequence of triangle A074909(reflected). [From Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Apr 10 2009]
%C A040027 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 
               2009: (Start)
%C A040027 The divergent series g(x=1,m) = 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ... 
               , m=>-1, is related to the sequence given above. For m=-1 this series 
               dates back to Euler. We discovered that g(x=1,m) = (-1)^m * (A040027(m) 
               - A000110(m+1) * A073003) with A073003 Gompertz's constant and A000110 
               the Bell numbers, see A163940; A040027(m = -1) = 0.
%C A040027 (End)
%H A040027 H. W. Gould and Jocelyn Quaintance, <a href="http://pefmath.etf.bg.ac.yu/
               vol1num2/AADM-Vol1-No2-371-385.pdf">A linear binomial recurrence 
               and the Bell numbers and polynomials</a>. Applicable Analysis and 
               Discrete Mathematics, 1 (2007), 371-385.
%H A040027 S. Kitaev, <a href="http://www.mat.univie.ac.at/users/slc/public_html/
               wpapers/s48kitaev.html">Generalized pattern avoidance with additional 
               restrictions</a>, Sem. Lothar. Combinat. B48e (2003).
%H A040027 S. Kitaev and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0205182">
               Simultaneous avoidance of generalized patterns</a>.
%F A040027 a(n) = b(n-2), n>1, b(n) = Sum_{k = 1..n} binomial(n, k-1)*b(n-k), b(0) 
               = 1. - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 28 2001
%F A040027 E.g.f. satisfies A'(x) = exp(x)*A(x)+1 [ N. J. A. Sloane (njas(AT)research.att.com) 
               ]
%F A040027 With offset 0, e.g.f.: x + exp(exp(x)) * int[0..x, t*exp(-exp(t)+t) dt] 
               (fits the recurrence up to n=215). - Ralf Stephan, Apr 25 2004
%F A040027 Recurrence : a(1)=1, a(2)=1, for n>2, a(n)=n-1+sum(j=2, n-1, binomial(n-1, 
               j)*a(j)) [gives a(n+1)] - Jon Perry (perry(AT)globalnet.co.uk), Apr 
               26 2005
%p A040027 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 
               2009: (Start)
%p A040027 nmax:=22; a(0):=1: for n from 1 to nmax do a(n):=sum(binomial(n, k-1)*a(n-k), 
               k = 1..n) od: seq(a(n),n=0..nmax);
%p A040027 (End)
%Y A040027 Left-hand border of triangle A046936.
%Y A040027 A153868 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 03 2009]
%Y A040027 A074909 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]
%Y A040027 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 
               2009: (Start)
%Y A040027 Row sums of A163940.
%Y A040027 (End)
%Y A040027 Sequence in context: A066571 A087648 A086616 this_sequence A071603 A090595 
               A027040
%Y A040027 Adjacent sequences: A040024 A040025 A040026 this_sequence A040028 A040029 
               A040030
%K A040027 easy,nonn,nice
%O A040027 0,3
%A A040027 H. W. Gould (gould(AT)math.wvu.edu)
%E A040027 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Dec 11 2006
%E A040027 Gould reference updated by Johannes W. Meijer (meijgia(AT)hotmail.com), 
               Aug 02 2009