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A156017
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Schroeder paths with two rise colors and two level colors.
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4
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1, 4, 24, 176, 1440, 12608, 115584, 1095424, 10646016, 105522176, 1062623232, 10840977408, 111811534848, 1163909087232, 12212421230592, 129027376349184, 1371482141884416, 14656212306231296, 157369985643577344, 1696975718802522112, 18369603773021552640
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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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.
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A059435, A090181.
Partial sums of A336283.
Sequence in context: A032349 A215709 A103334 * A000309 A112914 A308543
Adjacent sequences: A156014 A156015 A156016 * A156018 A156019 A156020
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry, Feb 01 2009
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EXTENSIONS
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Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010
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STATUS
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approved
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