A126079 -id:A126079 - cp's OEIS frontend

A070031 Expansion of (1+xC)C^3, where C = (1-sqrt(1-4x))/(2x) 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

N. J. A. Sloane, Jun 06 2002

nonn

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

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.

Cf. A000245, A000108, A126079, A306423.

Partial sums of A071736.

[2*(5*n+3)* Binomial(2*n+1, n)/((n+2)*(n+3)): n in [0..30]]; // G. C. Greubel, Feb 14 2019

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

Table[2^(n + 1)*(5*n + 3)*(2*n + 1)!!/(n + 3)!, {n, 0, 27}] (* Jean-François Alcover, Nov 07 2016 *)

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

((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

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)

A320825 Expansion of -(10x^2 - 6x + 1)sqrt(1 - 4x)/(3*x - 1)^2.

Original entry on oeis.org

-1, 2, 1, 0, -3, -10, -17, 28, 435, 2710, 13489, 60392, 254211, 1028250, 4046977, 15621932, 59463209, 224047866, 838012755, 3118339056, 11563677321, 42790868982, 158180470803, 584617335420, 2161733579313, 8001589660746, 29660171058675, 110136151678696, 409773163539325

Offset: 0

Peter Luschny, Oct 22 2018

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A126079, A320826, A320827.

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(-(10*x^2-6*x+1)*Sqrt(1-4*x)/(1-3*x)^2)); // G. C. Greubel, Oct 27 2018

c := n -> catalan(n)*(n-3)*(n+1)/((2*n-3)*(2*n-1)):
h := n -> hypergeom([1, -n], [5/2 - n], 3/4): A320825 := n -> c(n)*h(n):
seq(simplify(A320825(n)), n=0..28);

CoefficientList[Series[-(10x^2 - 6x + 1) Sqrt[1 - 4x]/(3x - 1)^2, {x, 0, 28}], x]

x='x+O('x^40); Vec(-(10*x^2-6*x+1)*sqrt(1-4*x)/(1-3*x)^2) \\ G. C. Greubel, Oct 27 2018

a(n) = c(n)*h(n) where c(n) = Catalan(n)*((n-3)*(n+1))/((2*n-3)*(2*n-1)) and h(n) = hypergeom([1, -n], [5/2 - n], 3/4).
A320825(n) = A320826(n) + A320827(n).
D-finite with recurrence: n*a(n) +(-13*n+15)*a(n-1) +4*(16*n-37)*a(n-2) +2*(-71*n+245)*a(n-3) +60*(2*n-9)*a(n-4)=0. - R. J. Mathar, Jan 23 2020

A126089 Expansion of e.g.f.: (1-2x)sqrt(1-4*x).

Original entry on oeis.org

1, -4, 4, 0, -48, -960, -20160, -483840, -13305600, -415134720, -14529715200, -564583219200, -24135932620800, -1126343522304000, -56992982228582400, -3108708121559040000, -181859425111203840000, -11359219476176732160000, -754576722346025779200000

Offset: 0

N. J. A. Sloane, Mar 22 2007

sign

Cf. A126967, A126079.

m:=20; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1-2*x)*Sqrt(1-4*x))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Jan 25 2020

CoefficientList[Series[(1-2*x)*Sqrt[1-4*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 02 2013 *)

seq(n)={Vec(serlaplace((1-2*x)*sqrt(1-4*x + O(x*x^n))))} \\ Andrew Howroyd, Jan 25 2020

a(n) ~ -2^(2*n-3/2)*n^(n-1)/exp(n). - Vaclav Kotesovec, Jun 02 2013
D-finite with recurrence: a(n) -4*n*a(n-1) +12*(2*n-7)*a(n-2)=0. - R. J. Mathar, Jan 24 2020
Conjecture D-finite with recurrence: (-n+4)*a(n) +2*(2*n-5)*(n-3)*a(n-1)=0. - R. J. Mathar, Jan 24 2020

cp's OEIS Frontend

A070031 Expansion of (1+xC)C^3, where C = (1-sqrt(1-4x))/(2x) is g.f. for Catalan numbers, A000108.

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A320825 Expansion of -(10x^2 - 6x + 1)sqrt(1 - 4x)/(3*x - 1)^2.

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A126089 Expansion of e.g.f.: (1-2x)sqrt(1-4*x).

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cp's OEIS Frontend

A070031 Expansion of (1+x*C)*C^3, where C = (1-sqrt(1-4*x))/(2*x) is g.f. for Catalan numbers, A000108.

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A320825 Expansion of -(10*x^2 - 6*x + 1)*sqrt(1 - 4*x)/(3*x - 1)^2.

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A126089 Expansion of e.g.f.: (1-2*x)*sqrt(1-4*x).

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A070031 Expansion of (1+xC)C^3, where C = (1-sqrt(1-4x))/(2x) is g.f. for Catalan numbers, A000108.

A320825 Expansion of -(10x^2 - 6x + 1)sqrt(1 - 4x)/(3*x - 1)^2.

A126089 Expansion of e.g.f.: (1-2x)sqrt(1-4*x).