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 A055879 Least nondecreasing sequence with a(1) = 1 and Hankel transform {1,1,1,1,...}. 9
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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 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 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. 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 Conjecture: (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 EXAMPLE 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 + ... 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 *) PROG (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 */ CROSSREFS Cf. A007317, A039658. Sequence in context: A245845 A097438 A205482 * A056470 A056471 A164904 Adjacent sequences:  A055876 A055877 A055878 * A055880 A055881 A055882 KEYWORD nonn AUTHOR John W. Layman, Jul 15 2000 EXTENSIONS More terms from Vincenzo Librandi, Feb 14 2014 STATUS approved

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Last modified October 20 12:27 EDT 2020. Contains 337904 sequences. (Running on oeis4.)