A156017 Schroeder paths with two rise colors and two level colors.

1, 4, 24, 176, 1440, 12608, 115584, 1095424, 10646016, 105522176, 1062623232, 10840977408, 111811534848, 1163909087232, 12212421230592, 129027376349184, 1371482141884416, 14656212306231296, 157369985643577344, 1696975718802522112, 18369603773021552640

Offset: 0

Author

Paul Barry, Feb 01 2009

Keywords

• easy
• nonn

Comments

Hankel transform is 8^C(n+1,2). - Philippe Deléham, Feb 04 2009

a(n-1) is also the number of ways a list of n items can be grouped into nested sublists (e.g., [a b c] to [a b c], [[a] b c], [[a, b] c], [[a [b]] c], and so on). - Ryan Tosh, Nov 10 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• 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.
• F. Chapoton, F. Hivert, and J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
• Z. Chen and H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO], 2016. See eq (1.13) a=4, b=2.
• Loïc Foissy, Generalized associative algebras, hal-03187479 [math.RA], 2021.

Crossrefs

Cf. A059435, A090181.

Partial sums of A336283.

Programs

Maple

A156017_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 2*(a[w-1]+add(a[j]*a[w-j-1], j=0..w-1)) od;
convert(a, list) end: A156017_list(20); # Peter Luschny, Feb 29 2016

Mathematica

CoefficientList[Series[(1-2*x-Sqrt[1-12*x+4*x^2])/(4*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
a[n_] := 2^n Hypergeometric2F1[- n, n + 1, 2, -1];
Table[a[n], {n, 0, 20}] (* Peter Luschny, Nov 25 2020 *)

Formula

G.f.: (1-2x-sqrt(1-12x+4x^2))/(4x);

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

a(n) = 2^n*Sum_{k=0..n} C(n+k,2k)*A000108(k) = 2^n*A006318(n).

D-finite with recurrence (n+1)*a(n) +6*(1-2*n)*a(n-1) +4*(n-2)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011

a(n) = Sum_{k=0..n} A090181(n,k)*2^(n+k). - Philippe Deléham, Nov 27 2011

a(n) ~ sqrt(4+3*sqrt(2))*(6+4*sqrt(2))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012

G.f.: 1/Q(0) where Q(k) = 1 + k*(1-2*x) - 2*x - 2*x*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013

a(n) = 2*A059435(n) for n >= 1. - Sergey Kirgizov, Feb 13 2017

a(n) = 2^n*hypergeom([-n, n + 1], [2], -1). - Peter Luschny, Nov 25 2020

Extensions

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010