%I A081696
%S A081696 1,1,3,9,29,97,333,1165,4135,14845,53791,196417,721887,2667941,9907851,
%T A081696 36950465,138320021,519515209,1957091277,7392602917,27992976565,
%U A081696 106236268337,404005515873,1539293204549,5875059106769,22459721336977
%N A081696 Expansion of 1/(x+Sqrt(1-4x)).
%C A081696 Number of irreducible ordered pairs of compositions of n. A pair of compositions 
               of n into the same number of (positive) parts, say n=a1+...+ak and 
               n=b1+...+bk, is irreducible if for all j<k, a1+...+aj is not equal 
               to b1+...+bj. E.g. a(3)=3 because the irreducible pairs are (1+2,
               2+1), (2+1,1+2), (3,3). - Herbert S. Wilf (wilf(AT)math.upenn.edu), 
               May 22 2004
%C A081696 Hankel transform is 2^n. - Paul Barry (pbarry(AT)wit.ie), Nov 22 2007
%C A081696 Equals left border of triangle A152229 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 29 2008]
%C A081696 Equals INVERTi transform of A000984: (1, 2, 6, 20, 70, 252,...). [From 
               Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]
%D A081696 Edward A. Bender, Gregory F. Lawler, Robin Pemantle and Herbert S. Wilf, 
               Irreducible compositions and the first return to the origin of a 
               random walk, preprint 2004
%D A081696 Paul Barry, A Catalan Transform and Related Transformations on Integer 
               Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H A081696 Edward A. Bender, Gregory F. Lawler, Robin Pemantle and Herbert S. Wilf, 
               <a href="http://front.math.ucdavis.edu/math.CO/0404253">Irreducible 
               compositions and the first return to the origin of a random walk</
               a>
%F A081696 G.f.: 1/(x+Sqrt[1-4x]). Recurrence: (n+3)y(n+3)-2(4n+9)y(n+2)+(15n+21)y(n+1)+2(2n+3)y(n) 
               = 0
%F A081696 A Catalan transform of the Fibonacci numbers F(n+1) under the mapping 
               G(x)-> G(xc(x)), c(x) the g.f. of A001008. The inverse mapping is 
               H(x)->H(x(1-x)). a(n)=sum{k=0..n, (k/(2n-k))binomial(2n-k, n-k)F(k+1)} 
               - Paul Barry (pbarry(AT)wit.ie), Dec 18 2004
%F A081696 G.f.: 1/(1-x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction). 
               [From Paul Barry (pbarry(AT)wit.ie), Aug 03 2009]
%t A081696 y[x_] := y[x] = (2(4n - 3)y[x - 1] - (15n - 24)y[x - 2] - (4n - 6)y[n 
               - 3])/n y[0] = 1 y[1] = 1 y[2] = 3
%Y A081696 Cf. A081698.
%Y A081696 Cf. A000045.
%Y A081696 A152229 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008]
%Y A081696 A000984 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]
%Y A081696 Sequence in context: A082306 A124431 A071740 this_sequence A148939 A077587 
               A001893
%Y A081696 Adjacent sequences: A081693 A081694 A081695 this_sequence A081697 A081698 
               A081699
%K A081696 easy,nonn
%O A081696 0,3
%A A081696 Emanuele Munarini (munarini(AT)mate.polimi.it), Apr 02 2003
%E A081696 More terms from Paul Barry (pbarry(AT)wit.ie), Dec 18 2004