0,3

Number of permutations beginning with 21 and avoiding 1-23. - Ralf Stephan, Apr 25 2004

Originally defined as main diagonal of an array of binomial recurrence coefficients (see Gould).

Starting (1, 3, 9, 31, 121,...) = row sums of triangle A153868 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 03 2009]

Equals eigensequence of triangle A074909(reflected). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

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.

(End)

H. W. Gould and Jocelyn Quaintance, A linear binomial recurrence and the Bell numbers and polynomials. Applicable Analysis and Discrete Mathematics, 1 (2007), 371-385.

S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).

S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns.

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

E.g.f. satisfies A'(x) = exp(x)*A(x)+1 [ N. J. A. Sloane (njas(AT)research.att.com) ]

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

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

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

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);

(End)

Left-hand border of triangle A046936.

A153868 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 03 2009]

A074909 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

Row sums of A163940.

(End)

Adjacent sequences: A040024 A040025 A040026 this_sequence A040028 A040029 A040030

Sequence in context: A066571 A087648 A086616 this_sequence A071603 A090595 A027040

easy,nonn,nice,new

H. W. Gould (gould(AT)math.wvu.edu)

Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Dec 11 2006

Gould reference updated by Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 02 2009