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A109983
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Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n, having k steps (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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1
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1, 0, 1, 2, 0, 0, 1, 6, 6, 0, 0, 0, 1, 12, 30, 20, 0, 0, 0, 0, 1, 20, 90, 140, 70, 0, 0, 0, 0, 0, 1, 30, 210, 560, 630, 252, 0, 0, 0, 0, 0, 0, 1, 42, 420, 1680, 3150, 2772, 924, 0, 0, 0, 0, 0, 0, 0, 1, 56, 756, 4200, 11550, 16632, 12012, 3432, 0, 0, 0, 0, 0, 0, 0, 0, 1, 72, 1260, 9240
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Row n has 2n+1 terms, the first n of which are 0. Row sums are the central Delannoy numbers (A001850). Column sums are the central trinomial coefficients (A002426) T(n,2n)=binomial(2n,n) (A000984). T(n,k)=A104684(n,2n-k). sum(k*T(n,k),k=0..n)=A109984(n)
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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
| T(n, k)=binomial(n, 2n-k)binomial(k, n). G :=1/sqrt[(1-tz)^2-4zt^2].
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EXAMPLE
| T(2,3)=6 because we have DNE, DEN, NED, END, NDE and EDN.
Triangle begins
.1;
.0,1,2;
.0,0,1,6,6;
.0,0,0,1,12,30,20;
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MAPLE
| T:=(n, k)->binomial(n, 2*n-k)*binomial(k, n):for n from 0 to 8 do seq(T(n, k), k=0..2*n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A001850, A002426, A000984, A104684, A109984.
Sequence in context: A138497 A113129 A127826 * A193033 A193531 A093492
Adjacent sequences: A109980 A109981 A109982 * A109984 A109985 A109986
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 07 2005
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