OFFSET
1,3
COMMENTS
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
From Petros Hadjicostas, Jan 15 2019: (Start)
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)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..200
B. N. Cyvin, E. Brendsdal, J. Brunvoll, and S. J. Cyvin, A class of polygonal systems representing polycyclic conjugated hydrocarbons: Catacondensed monoheptafusenes, 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)).
S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, and E. Brendsdal, Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
FORMULA
G.f.: 8*x^2*(1-x)/(1 - x + sqrt(1 - 6*x + 5*x^2))^3. - Emeric Deutsch, Oct 24 2002
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
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]
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
a(n) ~ 3*5^(n+1/2)/(8*sqrt(Pi)*n^(3/2)). - 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).
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Emeric Deutsch, Oct 24 2002
STATUS
approved