Search: id:A026671
Results 1-1 of 1 results found.

%I A026671
%S A026671 1,3,11,43,173,707,2917,12111,50503,211263,885831,3720995,15652239,
%T A026671 65913927,277822147,1171853635,4945846997,20884526283,88224662549,
%U A026671 372827899079,1576001732485,6663706588179,28181895551161,119208323665543
%N A026671 Number of lattice paths from (0,0) to (n,n), n >= 1, with steps (0,1), 
               (1,0) and, when on the diagonal, (1,1).
%C A026671 1, 1, 3, 11, 43, 173, ... is the unique sequence for which both the Hankel 
               transform of the sequence itself and the Hankel transform of its 
               left shift are the powers of 2 (A000079). For example, det[{{1, 1, 
               3}, {1, 3, 11}, {3, 11, 43}}] = det[{{1, 3, 11}, {3, 11, 43}, {11, 
               43, 173}}] = 4. - David Callan (callan(AT)stat.wisc.edu), Mar 30 
               2007
%C A026671 Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 25 2009: (Start)
%C A026671 a(n) is the image of F(2n+2) under the Catalan matrix (1,xc(x)) where 
               c(x) is the g.f. of A000108.
%C A026671 The sequence 1,1,3,... is the image of A001519 under (1,xc(x)). This 
               sequence has g.f. given by
%C A026671 1/(1-x-2x^2/(1-3x-x^2/(1-2x-x^2/(1-2x-x^2/(1-..... (continued fraction). 
               (End)
%C A026671 Binomial transform of A111961. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Feb 11 2009]
%H A026671 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 
               4 (2001), #01.1.5.
%H A026671 Miklos Bona, The permutation classes equinumerous to the smooth class, 
               <a href="http://www.combinatorics.org/">Electron. J. Combin.</a>, 
               5 (1998), no. 1, Research Paper 31, 12 pp.
%F A026671 G.f.: 1/(sqrt(1-4*x)-x); a(n)= sum(a(i-1)*binomial(2*(n-i), n-i), i=1..n) 
               + binomial(2*n, n), n >= 1, a(0)=1 - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Mar 21 2000
%F A026671 G.f.: 1/(1 -x -2*x*c(x)) where c(x) = g.f. for Catalan numbers A000108. 
               - Michael Somos Apr 20 2007
%F A026671 Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 25 2009: (Start)
%F A026671 G.f.: 1/(1-3xc(x)+x^2*c(x)^2);
%F A026671 G.f.: 1/(1-3x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-.... (continued fraction).
%F A026671 a(0)=1, a(n)=sum{k=0..n, (k/(2n-k))*C(2n-k,n-k)*F(2k+2)}. (End)
%F A026671 a(n)=Sum_{k, 0<=k<=n} A039599(n,k)*A000045(k+2). [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Feb 11 2009]
%F A026671 Contribution from Paul Barry (pbarry(AT)wit.ie), Feb 08 2009: (Start)
%F A026671 G.f.: 1/(1-x/(1-2x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-.... (continued fraction);
%F A026671 G.f. of 1,1,3,... is 1/(1-x-2x/(1-x/(1-x/(1-x/(1-.... (continued fraction). 
               (End)
%o A026671 (PARI) {a(n)= if(n<0, 0, polcoeff( 1/(sqrt(1 -4*x +x*O(x^n)) -x), n))} 
               /* Michael Somos Apr 20 2007 */
%Y A026671 a(n)=T(2n-1, n-1), T given by A026736, a(n)=T(2n, n), T given by A026670, 
               a(n)=T(2n+1, n+1), T given by A026725. Row sums of triangle A054335.
%Y A026671 Sequence in context: A140803 A084643 A007583 this_sequence A026876 A151090 
               A059278
%Y A026671 Adjacent sequences: A026668 A026669 A026670 this_sequence A026672 A026673 
               A026674
%K A026671 nonn,easy
%O A026671 0,2
%A A026671 Clark Kimberling (ck6(AT)evansville.edu); Miklos Bona (bona(AT)math.ufl.edu)

Search completed in 0.001 seconds