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Search: id:A110184
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| A110184 |
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Number of (1,1)-steps on the lines y=x, y=x+1, and y=x-1 in all Delannoy paths of length n (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
2
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| 0, 1, 8, 55, 354, 2205, 13484, 81523, 489158, 2919481, 17356752, 102884271, 608460330, 3591886293, 21172419444, 124649246955, 733107494286, 4307974826097, 25296523200920, 148448166035239, 870665283937010
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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a(n)=sum(k*A110183(n,k),k=0..n).
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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.=z(1-z-zr+z^2+z^2*r)/[(1-6z+z^2)(1-3z+z^2-zr+z^2*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)
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EXAMPLE
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a(3)=55 because on the 63 (=A001850(3)) Delannoy paths of length 3 we have alltogether A108666(3)=57 D-steps; however 2 of these, namely the D's in NNDEE and EEDNN, are not on the lines y=x, y=x+1, y=x-1.
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MAPLE
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r:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*(1-z-z*r+z^2+z^2*r)/(1-6*z+z^2)/(1-3*z+z^2-z*r+z^2*r): Gser:=series(G, z=0, 27): 0, seq(coeff(Gser, z^n), n=1..24);
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CROSSREFS
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Cf. A001850, A108666, A110183.
Sequence in context: A080312 A116885 A026994 this_sequence A013698 A075734 A033890
Adjacent sequences: A110181 A110182 A110183 this_sequence A110185 A110186 A110187
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 14 2005
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