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A039919
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Related to enumeration of edge-rooted catafusenes.
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7
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0, 1, 5, 21, 86, 355, 1488, 6335, 27352, 119547, 528045, 2353791, 10575810, 47849685, 217824285, 996999525, 4585548680, 21182609875, 98236853415, 457211008415, 2134851575050, 9997848660345, 46949087361550, 221022160284101, 1042916456739696, 4931673470809525, 23367060132453323
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OFFSET
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1,3
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COMMENTS
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Binomial transform of the first differences of the Catalan numbers (see A000245). - Paul Barry, Feb 16 2006
Starting (1, 5, 21, ...) = A002212, (1, 3, 10, 36, 137, ...) convolved with A007317, (1, 2, 5, 15, 51, ...). - Gary W. Adamson, May 19 2009
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.)
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.
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.
It turns out that M'(m) = a(floor((m + 1)/2)) for m >= 3, where (a(n): n >= 1) is the current sequence.
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).
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)).
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).
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).
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).
(End)
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LINKS
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FORMULA
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G.f.: 8*x^2*(1-x)/(1 - x + sqrt(1 - 6*x + 5*x^2))^3. - Emeric Deutsch, Oct 24 2002
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
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).
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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