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A055450
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Path-counting array T; each step of a path is (1 right) or (1 up) to a point below line y=x, else (1 right and 1 up) or (1 up) to a point on the line y=x, else (1 left) or (1 up) to a point above line y=x. T(i,j)=number of paths to point (i-j,j), for 1<=j<=i, i >= 1.
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7
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1, 1, 3, 1, 2, 10, 1, 3, 7, 36, 1, 4, 5, 26, 137, 1, 5, 9, 19, 101, 543, 1, 6, 14, 14, 75, 406, 2219, 1, 7, 20, 28, 56, 305, 1676, 9285, 1, 8, 27, 48, 42, 230, 1270, 7066, 39587, 1, 9, 35, 75, 90, 174, 965, 5390, 30302, 171369, 1, 10, 44, 110
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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FORMULA
| Initial values: T(i, 0)=1 for i >= 0. Recurrence: if 1<=j<i/2, then T(i, j)=T(i-1, j-1)+T(i-1, j); T(2j, j)=T(2j-2, j-1)+T(2j-1, j-1); else T(2j-k, j)=T(2j-k+1, j)+T(2j-k-1, j-1) for j=k, k+1, k+2, ..., for k=1, 2, 3, ...
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EXAMPLE
| T(4,4) defined as T(5,4)+T(3,3) when k=4, T(5,4) already defined when k=3.
Rows: {1}; {1,3}; {1,2,10}; {1,3,7,36}; {1,4,5,26,137}; ...
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CROSSREFS
| T(2n, n)=A000108(n) for n >= 0: Catalan numbers. T(n, n)=A002212(n), T(n, n-1)=A045868(n).
Partial diagonals: {1}, {1, 2}, {1, 3, 5}, {1, 4, 9, 14}, ... form A030237.
Sequence in context: A058142 A058144 A132323 * A185835 A126226 A144156
Adjacent sequences: A055447 A055448 A055449 * A055451 A055452 A055453
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KEYWORD
| nonn,tabl
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), May 18 2000
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