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Search: id:A118964
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| A118964 |
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Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that have k double rises above the x-axis (n>=1,k>=0). (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1); a double rise in a Grand Dyck path is an occurrence of uu in the path.) |
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2
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| 2, 5, 1, 14, 5, 1, 42, 19, 8, 1, 132, 67, 40, 12, 1, 429, 232, 166, 79, 17, 1, 1430, 804, 634, 395, 145, 23, 1, 4862, 2806, 2335, 1708, 879, 249, 30, 1, 16796, 9878, 8480, 6824, 4376, 1823, 404, 38, 1, 58786, 35072, 30691, 26137, 19334, 10521, 3542, 625, 47, 1
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Row sums are the central binomial coefficients (A000984). T(n,0)=A000108(n+1) (the Catalan numbers). T(n,1)=A114277(n-2). Sum(k*T(n,k),k>=0)=A000531(n-1). For all double rises (above, below and on the x-axis), see A118963.
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FORMULA
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G.f.=G(t,z)=(1+r)/[1-z(1+r)C]-1, where r=r(t,z) is the Narayana function, defined by (1+r)(1+tr)z=r, r(t,0)=0 and C=C(z)=[1-sqrt(1-4z)]/(2z) is the Catalan function. More generally, the g.f. H=H(t,s,u,z), where t,s and u mark double rises above, below and on the x-axis, respectively, is H=[1+r(s,z)]/[1-z(1+tr(t,z))(1+ur(s,z))].
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EXAMPLE
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T(3,1)=5 because we have u/ududd,u/uddud,udu/udd,duu/udd and u/udddu (the double rises above the x-axis are indicated by /.
Triangle starts:
2;
5,1;
14,5,1;
42,19,8,1;
132,67,40,12,1;
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MAPLE
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C:=(1-sqrt(1-4*z))/2/z: r:=(1-z-t*z-sqrt(z^2*t^2-2*z^2*t-2*z*t+z^2-2*z+1))/2/t/z: G:=(1+r)/(1-z*C*(1+r))-1: Gser:=simplify(series(G, z=0, 15)): for n from 1 to 11 do P[n]:=coeff(Gser, z, n) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=0..n-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000984, A000108, A114277, A000531, A118963.
Sequence in context: A156067 A101920 A114494 this_sequence A073187 A138159 A118919
Adjacent sequences: A118961 A118962 A118963 this_sequence A118965 A118966 A118967
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), May 07 2006
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