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Search: id:A110171
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| A110171 |
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Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n that start with exactly k (0,1) steps (or, equivalently, with exactly k (1,0) steps; a Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1)). |
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+0
1
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| 1, 2, 1, 8, 4, 1, 38, 18, 6, 1, 192, 88, 32, 8, 1, 1002, 450, 170, 50, 10, 1, 5336, 2364, 912, 292, 72, 12, 1, 28814, 12642, 4942, 1666, 462, 98, 14, 1, 157184, 68464, 27008, 9424, 2816, 688, 128, 16, 1, 864146, 374274, 148626, 53154, 16722, 4482, 978, 162, 18, 1
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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Row sums are the central Delannoy numbers (A001850). T(n,0)=A002003(n) for n>=1. T(n,1)=A050146(n) for n>=1. Column k for k>=1 has g.f. z^k*R^(k-1)*g*(1+z*R), where R=1+zR+zR^2=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318) and g=1/sqrt(1-6z+z^2) is the g.f. of the central Delannoy numbers (A001850). sum(k*T(n,k),k=0..n)=A050151(n) (the partial sums of the central Delannoy numbers) = (1/2)*n*R(n), where R(n)=A006318(n) is the n-th large Schroeder number.
Contribution from Paul Barry (pbarry(AT)wit.ie), May 07 2009: (Start)
Riordan array ((1+x+sqrt(1-6x+x^2))/(2*sqrt(1-6x+x^2)),(1-x-sqrt(1-6x+x^2))/2).
Inverse of Riordan array ((1-2x-x^2)/(1-x^2),x(1-x)/(1+x)). (End)
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REFERENCES
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R. A. Sulanke, Objects counted by the central Delannoy numbers, J. of Integer Sequences, 6, 2003, Article 03.1.5.
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FORMULA
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G.f.=(1+z+Q)/[Q(2-t+tz+tQ)], where Q=sqrt(1-6z+z^2).
Number triangle T(n,k)=[x^(n-k)]((1+x)/(1-x))^n. [From Paul Barry (pbarry(AT)wit.ie), May 07 2009]
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EXAMPLE
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T(2,1)=4 because we have NED, NENE, NEEN and NDE.
Triangle starts:
1;
2,1;
8,4,1;
38,18,6,1;
192,88,32,8,1;
Contribution from Paul Barry (pbarry(AT)wit.ie), May 07 2009: (Start)
Production matrix is
2, 1,
4, 2, 1,
6, 2, 2, 1,
8, 2, 2, 2, 1,
10, 2, 2, 2, 2, 1,
12, 2, 2, 2, 2, 2, 1,
14, 2, 2, 2, 2, 2, 2, 1,
16, 2, 2, 2, 2, 2, 2, 2, 1,
18, 2, 2, 2, 2, 2, 2, 2, 2, 1 (End)
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MAPLE
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Q:=sqrt(1-6*z+z^2): G:=(1+z+Q)/Q/(2-t+t*z+t*Q): Gser:=simplify(series(G, z=0, 13)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 10 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001850, A002003, A050146, A006318, A050151.
Sequence in context: A154537 A110446 A109979 this_sequence A104988 A136225 A089460
Adjacent sequences: A110168 A110169 A110170 this_sequence A110172 A110173 A110174
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 14 2005
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