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A070031
Expansion of (1+x*C)*C^3, where C = (1-sqrt(1-4*x))/(2*x) is g.f. for Catalan numbers, A000108.
5
1, 4, 13, 42, 138, 462, 1573, 5434, 19006, 67184, 239666, 861764, 3120180, 11366370, 41630805, 153216570, 566343030, 2101610280, 7826451270, 29240172780, 109566326220, 411671536380, 1550629453698, 5854180360932, 22148866939948, 83965042615552, 318895250752708
OFFSET
0,2
COMMENTS
Counts the large components twice and the small components once in all Dyck (n+1)-paths, i.e., twice the number of returns less the number of hills = 2*A000245(n+1) - A000108(n+1). - David Scambler, Oct 08 2012
For n>=2, the number of coalescent histories for matching gene tree and species trees with a pseudocaterpillar shape that has n+3 leaves (Rosenberg 2007, Corollary 3.9). - Noah A Rosenberg, Feb 14 2019
LINKS
F. Disanto and N. A. Rosenberg, Asymptotic properties of the number of matching coalescent histories for caterpillar-like families of species trees, IEEE/ACM Trans. Comput. Biol. Bioinformat. 13 (2016), 913-925. See Eq. 30.
Manuel Flores, Yuta Kimura, and Baptiste Rognerud, Combinatorics of quasi-hereditary structures, arXiv:2004.04726 [math.RT], 2020.
N. A. Rosenberg, Counting coalescent histories, J. Comput. Biol. 14 (2007), 360-377.
FORMULA
a(n) = 2*(5*n+3)*binomial(2*n+1, n)/((n+2)*(n+3)). - Emeric Deutsch, Dec 13 2002
a(n) = leftmost term of M^n*V, M = an infinite tridiagonal matrix with all 1's in the super and subdiagonals and all 2's in the main diagonal; with the rest zeros. V = Vector [1,2,0,0,0,...]. - Gary W. Adamson, Jun 16 2011
a(n) = 2*A000245(n+1) - A000108(n+1). - David Scambler, Oct 08 2012
D-finite with recurrence: 2*(n+3)*a(n) +(-11*n-15)*a(n-1) +6*(2*n-1)*a(n-2)=0. - R. J. Mathar, Aug 25 2013
G.f.: (3-sqrt(1-4*x))*(1-sqrt(1-4*x))^3/(16*x^3). - G. C. Greubel, Feb 14 2019
From Mélika Tebni, Sep 03 2024: (Start)
a(n) = A000108(n+1) - A126079(n+3).
E.g.f.: 4*exp(2*x)*(BesselI(0, 2*x) - 3/(4*x)*BesselI(1, 2*x) - (1-1/x)*BesselI(2, 2*x)). (End)
MAPLE
gf := ((3*x - 2)*sqrt(1 - 4*x) + 2*x^2 - 7*x + 2)/(2*x^3): ser := series(gf, x, 32): seq(coeff(ser, x, n), n = 0..9); # Peter Luschny, Jun 17 2022
MATHEMATICA
Table[2^(n + 1)*(5*n + 3)*(2*n + 1)!!/(n + 3)!, {n, 0, 27}] (* Jean-François Alcover, Nov 07 2016 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((3-sqrt(1-4*x))*(1-sqrt(1-4*x))^3/(16*x^3)) \\ G. C. Greubel, Feb 14 2019
(Magma) [2*(5*n+3)* Binomial(2*n+1, n)/((n+2)*(n+3)): n in [0..30]]; // G. C. Greubel, Feb 14 2019
(Sage) ((3-sqrt(1-4*x))*(1-sqrt(1-4*x))^3/(16*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 14 2019
CROSSREFS
Partial sums of A071736.
Sequence in context: A274952 A010900 A175005 * A082989 A267240 A192802
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 06 2002
STATUS
approved