1,3

Hankel transform {t(n)} of {a(n)} is given by t(n) = Det[{a(1), a(2), ..., a(n)}, {a(2), a(3), ..., a(n+1)}, ..., {a(n), a(n+1), ..., a(2n-1)}].

The bisections of this sequence appear to be the binomial transform of the Catalan numbers, A007317. If that is true then the g.f. for this sequence is (1/(2*x))*( 1 + x - (1-x)^(-1)*(1-x^2)^(1/2)*(1-5*x^2)^(1/2)), which occurs in the Cyvin et al. reference.

Self-convolution yields A039658 (shifted left), which is related to enumeration of edge-rooted catafusenes. [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 08 2008]

B. N. Cyvin et al., A class of polygonal systems representing polycyclic conjugated hydrocarbons ..., Monat. f. Chemie, 125 (1994), 1327-1337 (see V(x)).

S. J. Cyvin et al., "Enumeration and classification of certain polygonal systems representing polycyclic conjugated hydrocarbons: annelated catafusenes", J. Chem. Inf. Comput. Sci., 34 (1994), 1174-1180. See Table 1 second column on page 1174.

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

G.f.: A(x) = sqrt( (1+x)*(1-3*x^2-sqrt(1-6*x^2+5*x^4))/(2*(1-x)) ). G.f. satisifes: A(x) = 1 + x*A(x) + x^2*A(x)*A(-x). [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 08 2008]

G.f.: 1/(1-x-x^2/(1+x-x^2/(1-x-x^2/(1+x-x^2/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Feb 11 2009]

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, B=subst(A, x, -x); A=1+x*A+x^2*A*B); polcoeff(A, n)} (PARI) {a(n)=polcoeff(sqrt((1+x)*(1-3*x^2-sqrt(1-6*x^2+5*x^4 +x^4*O(x^n)))/(2*(1-x))), n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 08 2008]

Sequence in context: A106181 A098887 A097438 this_sequence A056470 A056471 A164904

Adjacent sequences: A055876 A055877 A055878 this_sequence A055880 A055881 A055882

nonn

John W. Layman (layman(AT)math.vt.edu), Jul 15 2000