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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.

%I A070031 #64 Apr 05 2025 10:12:13
%S A070031 1,4,13,42,138,462,1573,5434,19006,67184,239666,861764,3120180,
%T A070031 11366370,41630805,153216570,566343030,2101610280,7826451270,
%U A070031 29240172780,109566326220,411671536380,1550629453698,5854180360932,22148866939948,83965042615552,318895250752708
%N A070031 Expansion of (1+x*C)*C^3, where C = (1-sqrt(1-4*x))/(2*x) is g.f. for Catalan numbers, A000108.
%C A070031 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
%C A070031 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
%H A070031 G. C. Greubel, <a href="/A070031/b070031.txt">Table of n, a(n) for n = 0..1000</a>
%H A070031 Filippo Disanto and Noah A. Rosenberg, <a href="https://doi.org/10.1109/TCBB.2015.2485217">Asymptotic properties of the number of matching coalescent histories for caterpillar-like families of species trees</a>, IEEE/ACM Trans. Comput. Biol. Bioinformat. 13 (2016), 913-925. See Eq. 30.
%H A070031 Manuel Flores, Yuta Kimura, and Baptiste Rognerud, <a href="https://arxiv.org/abs/2004.04726">Combinatorics of quasi-hereditary structures</a>, arXiv:2004.04726 [math.RT], 2020.
%H A070031 Anna Rodriguez Rasmussen, <a href="https://arxiv.org/abs/2504.01706">Exact Borel subalgebras of quasi-hereditary monomial algebras</a>, arXiv:2504.01706 [math.RT], 2025. See p. 38.
%H A070031 Noah A. Rosenberg, <a href="https://doi.org/10.1089/cmb.2006.0109">Counting coalescent histories</a>, J. Comput. Biol. 14 (2007), 360-377.
%F A070031 a(n) = 2*(5*n+3)*binomial(2*n+1, n)/((n+2)*(n+3)). - _Emeric Deutsch_, Dec 13 2002
%F A070031 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
%F A070031 a(n) = 2*A000245(n+1) - A000108(n+1). - _David Scambler_, Oct 08 2012
%F A070031 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
%F A070031 G.f.: (3-sqrt(1-4*x))*(1-sqrt(1-4*x))^3/(16*x^3). - _G. C. Greubel_, Feb 14 2019
%F A070031 From _Mélika Tebni_, Sep 03 2024: (Start)
%F A070031 a(n) = A000108(n+1) - A126079(n+3).
%F A070031 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)
%p A070031 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
%t A070031 Table[2^(n + 1)*(5*n + 3)*(2*n + 1)!!/(n + 3)!, {n, 0, 27}] (* _Jean-François Alcover_, Nov 07 2016 *)
%o A070031 (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
%o A070031 (Magma) [2*(5*n+3)* Binomial(2*n+1, n)/((n+2)*(n+3)): n in [0..30]]; // _G. C. Greubel_, Feb 14 2019
%o A070031 (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
%Y A070031 Cf. A000245, A000108, A126079, A306423.
%Y A070031 Partial sums of A071736.
%K A070031 nonn
%O A070031 0,2
%A A070031 _N. J. A. Sloane_, Jun 06 2002