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A110170
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First differences of the central Delannoy numbers (A001850).
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4
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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, 6961037946938250, 39697830840765090, 226596964146630658
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OFFSET
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0,2
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COMMENTS
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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).
For n > 0: a(n) = A128966(2*n,n). - Reinhard Zumkeller, Jul 20 2013
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.
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FORMULA
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G.f.: (1-z)/sqrt(1-6*z+z^2).
a(n) = P_n(3)-P_{n-1}(3) (n>=1), where P_j is j-th Legendre polynomial.
From Paul Barry, 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)
Recurrence: n*(2*n-3)*a(n) = 2*(6*n^2-12*n+5)*a(n-1) - (n-2)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 2^(-1/4)*(3+2*sqrt(2))^n/sqrt(Pi*n). - Vaclav Kotesovec, Oct 18 2012
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MAPLE
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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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MATHEMATICA
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CoefficientList[Series[(1-x)/Sqrt[1-6*x+x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)
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PROG
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(PARI) x='x+O('x^66); Vec((1-x)/sqrt(1-6*x+x^2)) \\ Joerg Arndt, May 16 2013
(Haskell)
a110170 0 = 1
a110170 n = a128966 (2 * n) n -- Reinhard Zumkeller, Jul 20 2013
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CROSSREFS
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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
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nonn
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AUTHOR
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Emeric Deutsch, Jul 14 2005
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STATUS
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approved
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