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A055879
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Least non-decreasing sequence with a(1) = 1 and Hankel transform {1,1,1,1,...}.
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4
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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. [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 08 2008]
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REFERENCES
| 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.
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LINKS
| J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
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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). [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]
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EXAMPLE
| x + x^2 + 2*x^3 + 2*x^4 + 5*x^5 + 5*x^6 + 15*x^7 + 15*x^8 + 51*x^9 + ...
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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 *)
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PROG
| (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]
(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 */
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CROSSREFS
| Cf. A007317, A039658.
Sequence in context: A098887 A097438 A205482 * A056470 A056471 A164904
Adjacent sequences: A055876 A055877 A055878 * A055880 A055881 A055882
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KEYWORD
| nonn
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AUTHOR
| John W. Layman (layman(AT)math.vt.edu), Jul 15 2000
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