0,2

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

Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 25 2009: (Start)

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.

The sequence 1,1,3,... is the image of A001519 under (1,xc(x)). This sequence has g.f. given by

1/(1-x-2x^2/(1-3x-x^2/(1-2x-x^2/(1-2x-x^2/(1-..... (continued fraction). (End)

Binomial transform of A111961. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 11 2009]

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Miklos Bona, The permutation classes equinumerous to the smooth class, Electron. J. Combin., 5 (1998), no. 1, Research Paper 31, 12 pp.

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

G.f.: 1/(1 -x -2*x*c(x)) where c(x) = g.f. for Catalan numbers A000108. - Michael Somos Apr 20 2007

Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 25 2009: (Start)

G.f.: 1/(1-3xc(x)+x^2*c(x)^2);

G.f.: 1/(1-3x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-.... (continued fraction).

a(0)=1, a(n)=sum{k=0..n, (k/(2n-k))*C(2n-k,n-k)*F(2k+2)}. (End)

a(n)=Sum_{k, 0<=k<=n} A039599(n,k)*A000045(k+2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 11 2009]

Contribution from Paul Barry (pbarry(AT)wit.ie), Feb 08 2009: (Start)

G.f.: 1/(1-x/(1-2x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-.... (continued fraction);

G.f. of 1,1,3,... is 1/(1-x-2x/(1-x/(1-x/(1-x/(1-.... (continued fraction). (End)

(PARI) {a(n)= if(n<0, 0, polcoeff( 1/(sqrt(1 -4*x +x*O(x^n)) -x), n))} /* Michael Somos Apr 20 2007 */

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.

Sequence in context: A140803 A084643 A007583 this_sequence A026876 A151090 A059278

Adjacent sequences: A026668 A026669 A026670 this_sequence A026672 A026673 A026674

nonn,easy

Clark Kimberling (ck6(AT)evansville.edu); Miklos Bona (bona(AT)math.ufl.edu)