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A108666
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Number of (1,1)-steps 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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3
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0, 1, 8, 57, 384, 2505, 16008, 100849, 628736, 3888657, 23900040, 146146473, 889928064, 5399971161, 32668236552, 197123362785, 1186790473728, 7131032334369, 42773183020296, 256161548120857, 1531966218561920
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OFFSET
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0,3
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COMMENTS
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a(n) = sum(k=0..n, k*A104684(k) )
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REFERENCES
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Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, Arxiv preprint arXiv:1203.6792, 2012. - From N. J. A. Sloane, Oct 03 2012
R. A. Sulanke, Objects counted by the central Delannoy numbers, J. of Integer Sequences, 6, 2003, Article 03.1.5.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
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FORMULA
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a(n) = sum(k=1..n, k*binomial(n, k)*binomial(2*n-k, n) ).
G.f.: z*(1-z)/(1-6*z+z^2)^(3/2).
Recurrence: (n-1)*(2*n-3)*a(n) = 4*(3*n^2-6*n+2)*a(n-1) - (n-1)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ (3+2*sqrt(2))^n*sqrt(n)/(2^(7/4)*sqrt(Pi)). - Vaclav Kotesovec, Oct 18 2012
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EXAMPLE
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a(2)=8 because in the 13 (=A001850(2)) Delannoy paths of length 2, namely, DD, DNE,DEN,NED,END,NDE,EDN,NENE,NEEN,ENNE,ENEN,NNEE and EENN, we have a total of eight D steps.
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MAPLE
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a:=n->sum(k*binomial(n, k)*binomial(2*n-k, n), k=1..n): seq(a(n), n=0..24);
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MATHEMATICA
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CoefficientList[Series[x*(1-x)/(1-6*x+x^2)^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)
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CROSSREFS
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Cf. A001850, A104684.
Sequence in context: A143570 A096711 A079926 * A164031 A023000 A097114
Adjacent sequences: A108663 A108664 A108665 * A108667 A108668 A108669
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KEYWORD
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nonn,changed
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AUTHOR
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Emeric Deutsch, Jul 07 2005
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STATUS
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approved
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