A155867 A 'Morgan Voyce' transform of the large Schroeder numbers A006318.

1, 3, 13, 65, 355, 2061, 12501, 78323, 503033, 3294373, 21916883, 147708777, 1006330457, 6919474163, 47956087733, 334658965641, 2349535729811, 16583609673797, 117608812053277, 837626242775875, 5988634758319665

Offset: 0

Author

Paul Barry, Jan 29 2009

Keywords

• easy
• nonn

Comments

Image of A006318 under the Riordan array (1/(1-x), x/(1-x)^2).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.

Crossrefs

Cf. A006318, A085478.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-3*x+x^2 -Sqrt(1-10*x+19*x^2-10*x^3+x^4))/(2*x*(1-x)) )); // G. C. Greubel, Jun 09 2021

Mathematica

A006318[n_]:= 2*Hypergeometric2F1[-n+1, n+2, 2, -1];
A155867[n_]:= Sum[Binomial[n+j, 2*j]*A006318[j], {j,0,n}];
Table[A155867[n], {n, 0, 40}] (* G. C. Greubel, Jun 09 2021 *)

Sage

def A155867_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-3*x+x^2 -sqrt(1-10*x+19*x^2-10*x^3+x^4))/(2*x*(1-x)) ).list()
A155867_list(40) # G. C. Greubel, Jun 09 2021

Formula

G.f.: (1 - 3*x + x^2 - sqrt(1 - 10*x + 19*x^2 - 10*x^3 + x^4))/(2*x*(1-x)).

G.f.: 1/(1 -x -2*x/(1 -x -x/(1 -x -2*x/(1 -x -x/(1 -x -2*x/(1 -x -x/(1 - ... (continued fraction).

a(n) = Sum_{k=0..n} binomial(n+k,2k)*A006318(k).

a(n) = Sum_{k=0..n} A085478(n,k)*A006318(k). - Philippe Deléham, Jan 31 2009

Conjecture: (n+1)*a(n) + (4-11*n)*a(n-1) + (29*n-43)*a(n-2) +(73-29*n)*a(n-3) + (11*n-40)*a(n-4) + (5-n)*a(n-5) = 0. - R. J. Mathar, Jul 24 2012

The above recurrence follows from the differential equation (4*x^4 - 14*x^3 + 15*x^2 - 7*x + 1)*A(x) - (x^6 - 11*x^5 + 29*x^4 - 29*x^3 + 11*x^2 - x)*A'(x) + x^4 - x^3 + x - 1 = 0 satisfied by the g.f. A(x). - Peter Bala, Sep 15 2024