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A047749
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If n=2m then C(3m,m)/(2m+1); if n=2m+1 then C(3m+1,m+1)/(2m+1).
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22
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1, 1, 1, 2, 3, 7, 12, 30, 55, 143, 273, 728, 1428, 3876, 7752, 21318, 43263, 120175, 246675, 690690, 1430715, 4032015, 8414640, 23841480, 50067108, 142498692, 300830572, 859515920, 1822766520, 5225264024, 11124755664, 31983672534
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Hankel transform appears to be a signed aerated version of A059492. - Paul Barry (pbarry(AT)wit.ie), Apr 16 2008
Row sums of inverse Riordan array (1, x(1-x^2))^(-1). - Paul Barry (pbarry(AT)wit.ie), Apr 16 2008
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REFERENCES
| L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
Bousquet, Michel; and Lamathe, Cedric; On symmetric structures of order two. Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176.
S. J. Cyvin et al., Enumeration of staggered conformers of alkanes: complete solution ..., J. Molec. Struct., 413 (1997), 237-239.
S. J. Cyvin et al., Enumeration of staggered conformers of alkanes..., J. Molec. Struct., 445 (1998), 127-13.
E. Deutsch, Problem 10751, Amer. Math. Monthly, 108 (Nov., 2001), 872-873.
E. Deutsch, S. Feretic and M. Noy, Diagonally convex directed polyominoes and even trees: a bijection and related issues, Discrete Math., 256 (2002), 645-654.
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LINKS
| M. Bousquet and C. Lamathe, Enumeration of solid trees according to edge number and edge degree distribution, Discr. Math., 298 (2005), 115-141.
Alexander Burstein, Sergi Elizalde and Toufik Mansour, Restricted Dumont permutations, Dyck paths and noncrossing partitions, arXiv math.CO/0610234. [Theorem 3.5]
C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
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FORMULA
| G.f. is 1+Z, where Z satisfies x*Z^3 + (3*x-2)*Z^2 + (3*x-1)*Z + x = 0. Equivalently, the g.f. Y satisfies x*Y^3 - 2*Y^2 + 3*Y - 1 = 0. - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 06 2002
Reversion of g.f. (x-2x^2)/(1-x)^3 (ignoring signs). - R. Stephan, Mar 22 2004
G.f.: (4/(3x))(sin((1/3)*asin(sqrt(27x^2/4))))^2+(2/sqrt(3x^2))*sin((1/3)*asin(sqrt(27x^2/4))); - Paul Barry (pbarry(AT)wit.ie), Nov 08 2006
G.f.: 1/(1-2*sin(asin(3*sqrt(3)x/2)/3)/sqrt(3)); - Paul Barry (pbarry(AT)wit.ie), Apr 16 2008
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Sep 20 2009: (Start)
G.f. satisfies: A(x) = 1 + x*A(x)^2*A(-x);
also, A(x)*A(-x) = B(x^2) where B(x) = 1 + x*B(x)^3 = g.f. of A001764.
(End)
G.f.: 1/(1-C(x)) where C(x) = Reverse(x-x^3) = x +x^3 +3*x^5 +12*x^7 +55*x^9 +... (cf. A001764). [Joerg Arndt, Apr 16 2011]
a(n) = upper left term in M^n, M = the infinite square production matrix:
1, 1, 0, 0, 0, 0,...
0, 0, 1, 0, 0, 0,...
1, 1, 0, 1, 0, 0,...
0, 0, 1, 0, 1, 0,...
1, 1, 0, 1, 0, 1,...
...
- Gary W. Adamson, Jul 14 2011
Conjecture: 8*n*(n+1)*a(n) +36*n*(n-2)*a(n-1) -6*(9n^2-18n+14)*a(n-2) -27*(3n-7)*(3n-8)*a(n-3)=0. - R. J. Mathar, Dec 19 2011
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MAPLE
| A047749 := proc(m) if m mod 2 = 1 then x := (m-1)/2; RETURN((3*x+1)!/((x+1)!*(2*x+1)!)); fi; x := m/2; RETURN((3*x)!/(x!*(2*x+1)!)); end;
A047749 := proc(m) local x; if m mod 2 = 1 then x := (m-1)/2; RETURN((3*x+1)!/((x+1)!*(2*x+1)!)); fi; RETURN(A001764(m/2)); end;
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PROG
| (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^2*subst(A, x, -x+x*O(x^n))); polcoeff(A, n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 20 2009]
(PARI) x='x+O('x^66); /* that many terms */
C(x)=serreverse(x-x^3); /* =x+x^3+3*x^5+12*x^7+55*x^9 +..., cf. A001764 */
s=1/(1-C(x)); /* g.f. */
Vec(s) /* show terms */ /* Joerg Arndt, Apr 16 2011 */
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CROSSREFS
| Cf. A001764.
Sequence in context: A089324 A111759 * A134565 A100982 A186009 A034786
Adjacent sequences: A047746 A047747 A047748 * A047750 A047751 A047752
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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