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Search: id:A109979
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| A109979 |
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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 (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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+0
2
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| 1, 2, 1, 8, 4, 1, 36, 20, 6, 1, 172, 104, 36, 8, 1, 852, 552, 212, 56, 10, 1, 4324, 2968, 1236, 368, 80, 12, 1, 22332, 16104, 7164, 2336, 580, 108, 14, 1, 116876, 87976, 41372, 14512, 3980, 856, 140, 16, 1, 618084, 483192, 238356, 88848, 26372, 6312, 1204, 176
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OFFSET
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0,2
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COMMENT
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Row sums are the central Delannoy numbers (A001850). First column yields A109980 sum(k*T(n,k),k=0..n)=A001109(n).
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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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G.f.=[tz-z+sqrt(1-6z+z^2)]/(1-6z+2tz^2-t^2*z^2).
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EXAMPLE
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T(2,1)=4 because we have DNE, DEN, NED, and END.
Triangle begins:
1;
2,1;
8,4,1;
36,20,6,1;
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MAPLE
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G:=(t*z-z+sqrt(1-6*z+z^2))/(1-6*z+2*t*z^2-t^2*z^2): Gser:=simplify(series(G, z=0, 13)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 10 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001850, A109980, A001109.
Sequence in context: A133214 A110107 A110446 this_sequence A110171 A104988 A136225
Adjacent sequences: A109976 A109977 A109978 this_sequence A109980 A109981 A109982
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 06 2005
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