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

0,2

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.

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

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 *)

Prog

(Magma) R<x>:=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
(Sage)
def A155867_list(prec):
    P.<x> = 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

Crossrefs

Cf. A006318, A085478.

Sequence in context: A141342 A232222 A241598 * A378257 A009102 A080227

Adjacent sequences: A155864 A155865 A155866 * A155868 A155869 A155870

Keyword

easy,nonn,changed

Author

Paul Barry, Jan 29 2009

Status

approved