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Search: id:A110438
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| A110438 |
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Triangular array giving the number of NSEW unit step lattice paths of length n with terminal height k subject to the following restrictions. The paths start at the origin (0,0) and take unit steps (0,1)=N(north), (0,-1)=S(south), (1,0)=E(east) and (-1,0)=W(west) such that no paths pass below the x-axis, no paths begin with W, all W steps remain on the x-axis, and there are no NS steps. |
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+0
1
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| 1, 1, 1, 2, 2, 1, 5, 4, 3, 1, 12, 10, 7, 4, 1, 29, 25, 18, 11, 5, 1, 71, 62, 47, 30, 16, 6, 1, 175, 155, 121, 82, 47, 22, 7, 1, 434, 389, 311, 220, 135, 70, 29, 8, 1, 1082, 979, 799, 584, 378, 212, 100, 37, 9, 1
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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The row sums are the even-indexed Fibonacci numbers.
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REFERENCES
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Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
A. Nkwanta, A Riordan matrix approach to unifying a selected class of combinatorial arrays, Congressus Numerantium, 160 (2003), pp. 33-55.
A. Nkwanta, A note on Riordan matrices, Contemporary Mathematics Series, AMS, 252 (1999), pp. 99-107.
A. Nkwanta, Lattice paths, generating functions, and the Riordan group, Ph.D. Thesis, Howard University, Washington DC, 1997.
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FORMULA
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Recurrence is d(0, 0)= 1, d(1, 0)=1, d(n+1, 0) = 2*d(n, 0) + sum(d(n-j, j)j>=1, n>=1 for leftmost column and d(n+1, k) = d(n, k-1) + d(n, k) + sum(d(n-j, k+j)j>=1, n>=2, k>=1, and n>j; Riordan array d(n, k): (((1-z)/2z)*(sqrt(1+z+z^2)/sqrt(1-3z+z^2) -1), ((1-z+z^2)-sqrt(1-2z-z^2-2z^3+z^4)/2z)).
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EXAMPLE
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Triangle starts:
1;
1,1;
2,2,1;
5,4,3,1;
12,10,7,4,1;
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CROSSREFS
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Cf. A097724.
Sequence in context: A016538 A134226 A127742 this_sequence A121460 A105292 A125177
Adjacent sequences: A110435 A110436 A110437 this_sequence A110439 A110440 A110441
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Asamoah Nkwanta (Nkwanta(AT)jewel.morgan.edu), Aug 10 2005
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