%I A096654
%S A096654 1,2,8,38,222,1522,11986,106542,1054766,11506538,137119578,1772006854,
%T A096654 24681524038,368577425634,5874202721042,99515904921182,1785757627196766,
%U A096654 33835407673201882,675016383080377546,14143200407398386678
%N A096654 Denominators of self-convergents to 1/(e-2).
%C A096654 The self-continued fraction of r>0 is here introduced as the sequence 
               (b(0), b(1), b(2), ...) defined as follows: put r(0)=r, b(0)=[r(0)] 
               and for n>=1, put r(n)=b(n-1)/(r(n-1)-b(n-1)) and b(n)=[r(n)]. This 
               differs from simple continued fraction, for which r(n)=1/(r(n-1)-b(n-1)). 
               Now r=lim(p(n)/q(n)), where p(0)=b(1), q(0)=1, p(1)=b(0)(b(1)+1), 
               q(1)=b(1) and for n>=2, p(n)=b(n)*p(n-1)+b(n-1)*p(n-2), q(n)=b(n)*q(n-1)+b(n-1)*q(n-2); 
               p(0),p(1),... are the numerators of the self-convergents to r; q(0),
               q(1),... are the denominators of the self-convergents to r. Thus 
               A096654 is given by a(n)=(n+1)*a(n-1)+n*a(n-2), a(0)=1, a(1)=2.
%C A096654 Number of increasing runs of odd length in all permutations of [n+1]. 
               Example: a(2)=8 because we have (123),13(2),(3)12,(2)13,23(1),(3)(2)(1) 
               (the runs of odd length are shown between parentheses). - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Aug 29 2004
%F A096654 E.g.f.=[3-x-2(1+x)exp(-x)]/(1-x)^3. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Aug 29 2004
%e A096654 a(2)=q(2)=3*2+2*1=8, a(3)=q(3)=4*8+3*2=38. The convergents p(0)/q(0) 
               to p(4)/q(4) are 1/1, 3/2, 11/8, 53/38, 309/222.
%p A096654 G:=(3-x-2*(1+x)*exp(-x))/(1-x)^3: Gser:=series(G,x=0,22): 1,seq(n!*coeff(Gser,
               x^n),n=1..21);
%Y A096654 Cf. A000255, A096655, A096656, A096657.
%Y A096654 Sequence in context: A020031 A001340 A058786 this_sequence A060389 A101714 
               A077318
%Y A096654 Adjacent sequences: A096651 A096652 A096653 this_sequence A096655 A096656 
               A096657
%K A096654 nonn
%O A096654 0,2
%A A096654 Clark Kimberling (ck6(AT)evansville.edu), Jul 01 2004
%E A096654 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 29 2004