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A110170
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First differences of the central Delannoy numbers (A001850).
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1
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1, 2, 10, 50, 258, 1362, 7306, 39650, 217090, 1196834, 6634890, 36949266, 206549250, 1158337650, 6513914634, 36718533570, 207412854786, 1173779487810, 6653482333450, 37770112857074, 214694383882498, 1221832400430482
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of Delannoy paths of length n that do not start with a (1,1) step (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)). Example: a(1)=2 because we have NE and EN. Column 0 of A110169 (also nonzero entries in each column of A110169).
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REFERENCES
| 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
| G.f. = (1-z)/sqrt(1-6z+z^2). a(n)=P_n(3)-P_{n-1}(3) (n>=1), where P_j is j-th Legendre polynomial.
Contribution from Paul Barry (pbarry(AT)wit.ie), Oct 18 2009: (Start)
G.f.: (1-x)/(1-x-2x/(1-x-x/(1-x-x/(1-x-x/(1-... (continued fraction);
G.f.: 1/(1-2x/((1-x)^2-x/(1-x/((1-x)^2-x/(1-x/((1-x)^2-x/(1-... (continued fraction);
a(n)=sum{k=0..n, (0^(n+k)+C(n+k-1,2k-1))*C(2k,k)}=0^n+sum{k=0..n, C(n+k-1,2k-1)*C(2k,k)}. (End)
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MAPLE
| with(orthopoly): a:=proc(n) if n=0 then 1 else P(n, 3)-P(n-1, 3) fi end: seq(a(n), n=0..25);
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CROSSREFS
| Cf. A001850, A110169.
Sequence in context: A015949 A020699 A020729 * A026332 A027908 A206637
Adjacent sequences: A110167 A110168 A110169 * A110171 A110172 A110173
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 14 2005
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