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A110446
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Triangle of Delannoy paths counted by number of diagonal steps not preceded by an east step.
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2
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1, 2, 1, 8, 4, 1, 32, 24, 6, 1, 136, 128, 48, 8, 1, 592, 680, 320, 80, 10, 1, 2624, 3552, 2040, 640, 120, 12, 1, 11776, 18368, 12432, 4760, 1120, 168, 14, 1, 53344, 94208, 73472, 33152, 9520, 1792, 224, 16, 1, 243392, 480096, 423936, 220416, 74592, 17136
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OFFSET
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0,2
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COMMENTS
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T(n,k) = number of Delannoy paths (A001850) of steps east(E), north(N) and diagonal (D) (i.e., northeast) from (0,0) to (n,n) containing k Ds not preceded by an E.
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LINKS
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FORMULA
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G.f. G(z, t)=Sum_{n>=k>=0}T(n, k)*z^n*t^k is given by G(z, t)= (1 - z(4 + 2*t) - z^2(4 - 4*t - t^2))^(-1/2)
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EXAMPLE
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Table begins
\ k...0....1....2....3....4....
n\
0 |...1
1 |...2....1
2 |...8....4....1
3 |..32...24....6....1
4 |.136..128...48....8....1
5 |.592..680..320...80...10....1
The paths ENDD, NDDE, DEND, DNDE, DDEN, DDNE each have 2 Ds not preceded by an E,
and so T(3,2)=6.
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MATHEMATICA
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T[n_, k_] := SeriesCoefficient[(1-z(4 + 2*t) - z^2 (4 - 4*t - t^2))^(-1/2), {z, 0, n}, {t, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 08 2016 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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