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A098979
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Triangle read by rows: counts Motzkin paths by length of final descent.
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0
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1, 1, 1, 1, 2, 2, 4, 4, 1, 9, 9, 3, 21, 21, 8, 1, 51, 51, 21, 4, 127, 127, 55, 13, 1, 323, 323, 145, 39, 5, 835, 835, 385, 113, 19, 1, 2188, 2188, 1030, 322, 64, 6, 5798, 5798, 2775, 910, 203, 26, 1, 15511, 15511, 7525, 2562, 622, 97, 7, 41835, 41835, 20526, 7203
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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FORMULA
| G.f. ( (1 + x)*(1 - x*y) - (1 + x*y)(1 - 2*x - 3*x^2)^(1/2) )/( 2*x*(1 - y + x*y(1 + x*y)) ) = Sum_{n>=0, 1<=k<=n/2}T(n, k)x^n*y^k.
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MATHEMATICA
| Clear[a] a[0, 0]=1; a[1, 0]=1; a[2, 0]=a[2, 1]=1; a[n_, r_]/; r<0 := 0; a[n_, r_]/; n>=2 && r==0:= Sum[a[n-1, n-1-j], {j, n-1}]; a[n_, r_]/; n>=3 && r >= 1 := a[n, r] = Sum[a[n-2, n-2-j], {j, n-r-1}]+Sum[a[n-1, n-1-j], {j, n-r-1}]; Table[a[n, r], {n, 0, 10}, {r, 0, n/2}]
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CROSSREFS
| Row sums are the Motzkin numbers (A001006), as are columns k=0 and k=1 (apart from initial 1's).
Sequence in context: A071511 A119922 A097860 * A071928 A165207 A130501
Adjacent sequences: A098976 A098977 A098978 * A098980 A098981 A098982
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KEYWORD
| nonn
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AUTHOR
| David Callan (callan(AT)stat.wisc.edu), Oct 24 2004
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