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A109980
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Number of Delannoy paths of length n with no (1,1)-steps on the line y=x (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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5
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1, 2, 8, 36, 172, 852, 4324, 22332, 116876, 618084, 3296308, 17702412, 95627580, 519170004, 2830862532, 15494401116, 85091200620, 468692890308, 2588521289812, 14330490031020, 79509491551772, 442019710668852
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Equals left border of triangle A152250 and INVERTi transform of A001850, the Delannoy numbers: (1, 3, 13, 63, 321,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 30 2008]
Hankel transform is A036442. First column of Riordan array ((1-x)/(1+x), x/(1+3x+2x^2))^{-1}. [From Paul Barry (pbarry(AT)wit.ie), Apr 27 2009]
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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)].
Moment representation: a(n)=(1/pi)int(x^n*sqrt(-x^2+6x-1)/(x(6-x)),x,3-2*sqrt(2),3+2*sqrt(2))+0^n/3. [From Paul Barry (pbarry(AT)wit.ie), Apr 27 2009]
a(n) is the upper left term in M^n, M = an infinite square production matrix as follows:
2, 2, 0, 0, 0, 0,...
2, 1, 2, 0, 0, 0,...
2, 1, 1, 2, 0, 0,...
2, 1, 1, 1, 2, 0,...
2, 1, 1, 1, 1, 2,...
... - Gary W. Adamson, Aug 23 2011
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EXAMPLE
| a(2)=8 because we have NDE, EDN, NENE, NEEN, ENNE, ENEN, NNEE and EENN.
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MAPLE
| g:=1/(z+sqrt(1-6*z+z^2)): gser:=series(g, z=0, 28): 1, seq(coeff(gser, z^n), n=1..25);
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CROSSREFS
| First column of A109979.
A152250 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 30 2008]
Sequence in context: A089387 A206902 A084868 * A186338 A190862 A110837
Adjacent sequences: A109977 A109978 A109979 * A109981 A109982 A109983
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 06 2005
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