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Least nondecreasing sequence with a(1) = 1 and Hankel transform {1,1,1,1,...}.
15

%I #42 Dec 15 2020 09:10:41

%S 1,1,2,2,5,5,15,15,51,51,188,188,731,731,2950,2950,12235,12235,51822,

%T 51822,223191,223191,974427,974427,4302645,4302645,19181100,19181100,

%U 86211885,86211885,390248055,390248055,1777495635,1777495635,8140539950,8140539950

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

%C 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)}].

%C 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.

%C Self-convolution yields A039658 (shifted left), which is related to enumeration of edge-rooted catafusenes. - _Paul D. Hanna_, Aug 08 2008

%H Vincenzo Librandi, <a href="/A055879/b055879.txt">Table of n, a(n) for n = 1..200</a>

%H B. N. Cyvin et al., <a href="http://dx.doi.org/10.1007/BF00811082">A class of polygonal systems representing polycyclic conjugated hydrocarbons: Catacondensed monoheptafusenes</a>, Monat. f. Chemie, 125 (1994), 1327-1337 (see V(x)).

%H S. J. Cyvin et al, <a href="https://doi.org/10.1021/ci00009a021">Enumeration and classification of Benezenoid systmes. 32. Normal Perifuses with two internal vertices</a>, J. Chem. Inf. Comput. Sci. 32 (1992) 532-540, Table 1.

%H S. J. Cyvin et al., <a href="http://dx.doi.org/10.1021/ci00021a026">Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes</a>, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180. See Table 1 second column on page 1174.

%H J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.

%F 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

%F 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

%F 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

%F 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

%F 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

%F 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

%e 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 + ...

%t 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 *)

%t 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 *)

%o (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)

%o (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

%o (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 */

%Y Cf. A007317, A039658.

%K nonn

%O 1,3

%A _John W. Layman_, Jul 15 2000

%E More terms from _Vincenzo Librandi_, Feb 14 2014