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%I #24 May 14 2017 09:30:13
%S 1,1,2,5,16,57,224,934,4092,18581,86888,415856,2029160,10061161,
%T 50568680,257129888,1320619176,6842177174,35722456976,187772944964,
%U 992991472328,5279633960181,28208037066528,151373637844440,815568695756496,4410124252008112
%N First row of Modified Schroeder numbers for q=3 (A114292).
%C a(i) is the number of paths from (0,0) to (i,i) using steps of length (0,1), (1,0) and (1,1), not passing above the line y=x nor below the line y=x/2.
%H Alois P. Heinz, <a href="/A114296/b114296.txt">Table of n, a(n) for n = 0..1000</a>
%H C. Hanusa, <a href="http://people.qc.cuny.edu/faculty/christopher.hanusa/research/Documents/papers/Dissertation.pdf">A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows</a>, PhD Thesis, 2005, University of Washington, Seattle, USA.
%F a(n) ~ c * (3+2*sqrt(2))^n / n^(3/2), where c = 0.02741316010407391604887680145773... . - _Vaclav Kotesovec_, Sep 07 2014
%e The number of paths from (0,0) to (3,3) staying between the lines y=x and y=x/2 using steps of length (0,1), (1,0) and (1,1) is a(3)=5.
%p b:= proc(x, y) option remember; `if`(y>x or y<x/2, 0,
%p `if`(x=0, 1, b(x, y-1)+b(x-1, y)+b(x-1, y-1)))
%p end:
%p a:= n-> b(n, n):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 25 2013
%t b[x_, y_] := b[x, y] = If[y>x || y<x/2, 0, If[x == 0, 1, b[x, y-1] + b[x-1, y] + b[x-1, y-1]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 05 2015, after _Alois P. Heinz_ *)
%Y See also A112833-A112844 and A114292-A114299.
%Y Cf. A224776, A225041. - _Alois P. Heinz_, Apr 25 2013
%Y Cf. A286761.
%K nonn
%O 0,3
%A Christopher Hanusa (chanusa(AT)math.binghamton.edu), Nov 21 2005
%E Corrected by _Philippe Deléham_, Sep 04 2006
%E Extended beyond a(10) by _Alois P. Heinz_, Apr 25 2013
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