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%I #77 Jun 27 2022 01:17:57
%S 1,4,24,176,1440,12608,115584,1095424,10646016,105522176,1062623232,
%T 10840977408,111811534848,1163909087232,12212421230592,
%U 129027376349184,1371482141884416,14656212306231296,157369985643577344,1696975718802522112,18369603773021552640
%N Schroeder paths with two rise colors and two level colors.
%C Hankel transform is 8^C(n+1,2). - _Philippe Deléham_, Feb 04 2009
%C 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
%H Vincenzo Librandi, <a href="/A156017/b156017.txt">Table of n, a(n) for n = 0..200</a>
%H F. Chapoton, F. Hivert, and J.-C. Novelli, <a href="http://arxiv.org/abs/1307.0092">A set-operad of formal fractions and dendriform-like sub-operads</a>, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
%H Z. Chen and H. Pan, <a href="http://arxiv.org/abs/1608.02448">Identities involving weighted Catalan-Schroder and Motzkin Paths</a>, arXiv:1608.02448 [math.CO], 2016. See eq (1.13) a=4, b=2.
%H Loïc Foissy, <a href="https://hal.archives-ouvertes.fr/hal-03187479">Generalized associative algebras</a>, hal-03187479 [math.RA], 2021.
%F G.f.: (1-2x-sqrt(1-12x+4x^2))/(4x);
%F G.f.: 1/(1-2x-2x/(1-2x-2x/(1-2x-2x/(1-... (continued fraction);
%F a(n) = 2^n*Sum_{k=0..n} C(n+k,2k)*A000108(k) = 2^n*A006318(n).
%F 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
%F a(n) = Sum_{k=0..n} A090181(n,k)*2^(n+k). - _Philippe Deléham_, Nov 27 2011
%F a(n) ~ sqrt(4+3*sqrt(2))*(6+4*sqrt(2))^n/(2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 20 2012
%F 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
%F a(n) = 2*A059435(n) for n >= 1. - _Sergey Kirgizov_, Feb 13 2017
%F a(n) = 2^n*hypergeom([-n, n + 1], [2], -1). - _Peter Luschny_, Nov 25 2020
%p A156017_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p 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;
%p convert(a, list) end: A156017_list(20); # _Peter Luschny_, Feb 29 2016
%t CoefficientList[Series[(1-2*x-Sqrt[1-12*x+4*x^2])/(4*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%t a[n_] := 2^n Hypergeometric2F1[- n, n + 1, 2, -1];
%t Table[a[n], {n, 0, 20}] (* _Peter Luschny_, Nov 25 2020 *)
%Y Cf. A059435, A090181.
%Y Partial sums of A336283.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Feb 01 2009
%E Spelling/notation corrections by _Charles R Greathouse IV_, Mar 18 2010
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