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A039919 Related to enumeration of edge-rooted catafusenes. 7

%I #72 Jan 18 2019 04:40:24

%S 0,1,5,21,86,355,1488,6335,27352,119547,528045,2353791,10575810,

%T 47849685,217824285,996999525,4585548680,21182609875,98236853415,

%U 457211008415,2134851575050,9997848660345,46949087361550,221022160284101,1042916456739696,4931673470809525,23367060132453323

%N Related to enumeration of edge-rooted catafusenes.

%C Binomial transform of the first differences of the Catalan numbers (see A000245). - _Paul Barry_, Feb 16 2006

%C Starting (1, 5, 21, ...) = A002212, (1, 3, 10, 36, 137, ...) convolved with A007317, (1, 2, 5, 15, 51, ...). - _Gary W. Adamson_, May 19 2009

%C From _Petros Hadjicostas_, Jan 15 2019: (Start)

%C In Cyvin et al. (1992), sequence (N(m): m >= 1) = (A002212(m): m >= 1) is defined by eq. (1), p. 533. (We may let N(0) := A002212(0) = 1.)

%C In the same reference, sequence (M(m): m >= 1) is defined by eq. (13), p. 534. We have M(2*m) = M(2*m-1) = A007317(m) for m >= 1.

%C In the same reference, the sequence (M'(m): m >= 3) is defined by eq. (26), p. 535; see also Cyvin et al. (1994, Monatshefte fur Chemie), eq. 5, p. 1329. We have M'(m) = Sum_{1 <= i <= floor((m-1)/2)} N(i)*M(m-2*i) for m >= 3.

%C It turns out that M'(m) = a(floor((m + 1)/2)) for m >= 3, where (a(n): n >= 1) is the current sequence.

%C If 1 + U(x) = Sum_{n >= 0} N(n)*x^n = Sum_{n >= 0} A002212(n)*x^n, then the g.f. of the sequence (M(m): m >= 1) is V(x) = x*(1-x)^(-1)*(1 + U(x^2)). See eqs. 3 and 4, p. 1329, in Cyvin et al. (1994, Monatshefte fur Chemie).

%C Eq. 6 in the latter reference (pp. 1329-1330) states that the g.f. of the sequence (M'(m): m >= 3) is U(x^2)*V(x) = U(x^2)*x*(1-x)^(-1)*(1 + U(x^2)).

%C Since M'(m) = a(floor((m + 1)/2)) for m >= 3, the latter g.f. also equals (1 + x)*A(x^2)/x, where A(x) = Sum_{n >= 1} a(n)*x^n is the g.f. of the current sequence (given below by Emeric Deutsch).

%C Equating the two forms of the g.f. of the (M'(m): m >= 3), we get that A(x) = x*U(x)*(1 + U(x))/(1-x), where 1 + U(x) is the g.f. of A002212 (with U(0) = 0).

%C The sequence (M'(m): m >= 3) = (a(floor((m + 1)/2)): m >= 3) is used in the calculation of A026298 (= numbers of polyhexes of the class PF2 with three catafusenes annelated to pyrene).

%C (End)

%H Vincenzo Librandi, <a href="/A039919/b039919.txt">Table of n, a(n) for n = 1..200</a>

%H B. N. Cyvin, E. Brendsdal, J. Brunvoll, and S. J. Cyvin, <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 Eq. 6 for the g.f. of the sequence (M'(n): n >= 3) = (a(floor((m + 1)/2)): m >= 3)).

%H S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, <a href="http://dx.doi.org/10.1021/ci00009a021">Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices</a>, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.

%H S. J. Cyvin, B. N. Cyvin, J. Brunvoll, and E. Brendsdal, <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.

%F G.f.: 8*x^2*(1-x)/(1 - x + sqrt(1 - 6*x + 5*x^2))^3. - _Emeric Deutsch_, Oct 24 2002

%F a(n) = A002212(n) - Sum_{j=0..n-1} A002212(j). Example: a(5) = 137 - (1 + 1 + 3 + 10 + 36) = 86. - _Emeric Deutsch_, Jan 23 2004

%F a(n+1) = Sum_{k=0..n} C(n,k)*(C(k+1) - C(k)) for n >= 0, where C(k) = A000108(k). - _Paul Barry_, Feb 16 2006 [edited by _Petros Hadjicostas_, Jan 18 2019]

%F Recurrence: (n-2)*(n+1)*a(n) = 2*(n-1)*(3*n-4)*a(n-1) - 5*(n-2)*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 08 2012

%F a(n) ~ 3*5^(n+1/2)/(8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 08 2012

%F G.f.: x*U(x)*(1 + U(x))/(1-x), where 1 + U(x) is the g.f. of A002212 (using the notation in the two papers by Cyvin et al. published in 1994).

%t Table[SeriesCoefficient[8x^2*(1-x)/(1-x+Sqrt[1-6x+5x^2])^3,{x,0,n}],{n,1,23}] (* _Vaclav Kotesovec_, Oct 08 2012 *)

%o (PARI) x='x+O('x^66); concat([0],Vec(8*x^2*(1-x)/(1-x+sqrt(1-6*x+5*x^2))^3)) \\ _Joerg Arndt_, May 04 2013

%Y Cf. A002212, A026298.

%Y Cf. A007317. - _Gary W. Adamson_, May 19 2009

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_

%E More terms from _Emeric Deutsch_, Oct 24 2002

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