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Search: id:A109984
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| A109984 |
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Number of 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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+0
2
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| 0, 5, 44, 321, 2184, 14325, 91860, 580097, 3622928, 22437477, 138049020, 844881345, 5148375192, 31258302933, 189199514532, 1142148091905, 6878977097760, 41347348295877, 248082231062988, 1486116788646977
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OFFSET
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0,2
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COMMENT
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a(n)=sum(k*A109983(k),k=0..2n).
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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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a(n)=sum(k*binomial(n, 2n-k)*binomial(k, n), k=n..2n). G.f.=z*(5-z)/(1-6z+z^2)^(3/2).
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EXAMPLE
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a(1)=5 because in the 3 (=A001850(1)) Delannoy paths of length 1, namely D,NE,and EN, we have alltogether five steps.
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MAPLE
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a:=n->add(k*binomial(n, 2*n-k)*binomial(k, n), k=n..2*n): seq(a(n), n=0..23);
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CROSSREFS
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Cf. A109983, A001850.
Sequence in context: A071861 A128523 A068311 this_sequence A096355 A054766 A106273
Adjacent sequences: A109981 A109982 A109983 this_sequence A109985 A109986 A109987
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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 07 2005
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