A055879 Least nondecreasing sequence with a(1) = 1 and Hankel transform {1,1,1,1,...}.

1, 1, 2, 2, 5, 5, 15, 15, 51, 51, 188, 188, 731, 731, 2950, 2950, 12235, 12235, 51822, 51822, 223191, 223191, 974427, 974427, 4302645, 4302645, 19181100, 19181100, 86211885, 86211885, 390248055, 390248055, 1777495635, 1777495635, 8140539950, 8140539950

Offset: 1

Author

John W. Layman, Jul 15 2000

Keywords

• nonn

Comments

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. - Paul D. Hanna, Aug 08 2008

Examples

			G.f.: x + x^2 + 2*x^3 + 2*x^4 + 5*x^5 + 5*x^6 + 15*x^7 + 15*x^8 + 51*x^9 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• B. N. Cyvin et al., A class of polygonal systems representing polycyclic conjugated hydrocarbons: Catacondensed monoheptafusenes, Monat. f. Chemie, 125 (1994), 1327-1337 (see V(x)).
• S. J. Cyvin et al, Enumeration and classification of Benezenoid systmes. 32. Normal Perifuses with two internal vertices, J. Chem. Inf. Comput. Sci. 32 (1992) 532-540, Table 1.
• S. J. Cyvin et al., Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes, J. Chem. Inform. 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.

Crossrefs

Cf. A007317, A039658.

Programs

Mathematica

a[ n_] := If[ n < 1, 0, With[ {m = n - 1}, SeriesCoefficient[ Nest[ 1 / (1 - x - x^2 / (1 + x - x^2 #)) &, 1, Quotient[ m + 1, 2]], {x, 0, m}]]]; (* Michael Somos, Jul 01 2011 *)
CoefficientList[Series[Sqrt[(1 + x) (1 - 3 x^2 - Sqrt[1 - 6 x^2 + 5 x^4])/(2 (1 - x))]/x^2, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 14 2014 *)

PARI

a(n)=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)=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) \\ Paul D. Hanna, Aug 08 2008

PARI

{a(n) = local(A); if( n<1, 0, n--; A = O(x); for( k = 0, n\2, A = 1 / (1 - x - x^2 / (1 + x - x^2 * A))); polcoeff( A, n))}; /* Michael Somos, Jul 01 2011 */

Formula

G.f.: A(x) = sqrt( (1+x)*(1-3*x^2-sqrt(1-6*x^2+5*x^4))/(2*(1-x)) ). G.f. satisfies: A(x) = 1 + x*A(x) + x^2*A(x)*A(-x). - Paul D. Hanna, 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). - Paul Barry, Feb 11 2009

D-finite with recurrence (n+1)*a(n) - a(n-1) + (-6*n+11)*a(n-2) + 5*a(n-3) + 5*(n-4)*a(n-4) = 0. - R. J. Mathar, Nov 26 2012

G.f.: sqrt((1+x)*(1-3*x^2-sqrt(1-6*x^2+5*x^4))/(2*(1-x)))/x. - Vaclav Kotesovec, Feb 13 2014

a(n) ~ (5+sqrt(5) - (-1)^n*(5-sqrt(5))) * sqrt(2) * 5^(n/2) / (8 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014

a(n) = a(n-1) if n is even. a(n) = a(n-1)+A002212((n-1)/2) if n is odd. [Cyvin (1992) eq (14)] - R. J. Mathar, Dec 15 2020

Extensions

More terms from Vincenzo Librandi, Feb 14 2014