A070031 Expansion of (1+x*C)*C^3, where C = (1-sqrt(1-4*x))/(2*x) is g.f. for Catalan numbers, A000108.
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
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Filippo Disanto and Noah 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.
- Anna Rodriguez Rasmussen, Exact Borel subalgebras of quasi-hereditary monomial algebras, arXiv:2504.01706 [math.RT], 2025. See p. 38.
- Noah A. Rosenberg, Counting coalescent histories, J. Comput. Biol. 14 (2007), 360-377.
Programs
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Magma
[2*(5*n+3)* Binomial(2*n+1, n)/((n+2)*(n+3)): n in [0..30]]; // G. C. Greubel, Feb 14 2019
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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
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Mathematica
Table[2^(n + 1)*(5*n + 3)*(2*n + 1)!!/(n + 3)!, {n, 0, 27}] (* Jean-François Alcover, Nov 07 2016 *) -
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 -
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
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
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)
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)
Comments