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

0,2

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

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.

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).

Conjecture: (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

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 *)

Crossrefs

Cf. A059435, A090181.

Partial sums of A336283.

Sequence in context: A032349 A215709 A103334 * A000309 A112914 A308543

Adjacent sequences: A156014 A156015 A156016 * A156018 A156019 A156020

Keyword

easy,nonn

Author

Paul Barry, Feb 01 2009

Extensions

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

Status

approved