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Search: id:A104559
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| A104559 |
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Triangle, read by rows, of the number of left factors of peakless Motzkin paths of length n having k number of U's and D's (i.e. number of paths from (0,0) to the line x=n, consisting of steps U=(1,1), H=(1,0), D=(1,1), that never go below the x-axis and a U step is never followed by a D step). |
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+0
5
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| 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 6, 1, 1, 5, 16, 18, 9, 1, 1, 6, 25, 40, 36, 12, 1, 1, 7, 36, 75, 100, 60, 16, 1, 1, 8, 49, 126, 225, 200, 100, 20, 1, 1, 9, 64, 196, 441, 525, 400, 150, 25, 1, 1, 10, 81, 288, 784, 1176, 1225, 700, 225, 30, 1, 1, 11, 100, 405, 1296, 2352
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Row sums form A091964, the number of left factors of peakless Motzkin paths of length n.
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FORMULA
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G.f.: A(x, y) = 2/(1-x+x^2*y^2 - 2*x*y + sqrt((1-x+x^2*y^2)^2 - 4*x^2*y^2)) (due to Emeric Deutsch). T(n, k) = C(n-[k/2], [(k+1)/2])*C(n-[(k+1)/2], [k/2]) = A104557(n, k)/(n-k)!.
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EXAMPLE
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Triangle begins:
1;
1,1;
1,2,1;
1,3,4,1;
1,4,9,6,1;
1,5,16,18,9,1;
1,6,25,40,36,12,1;
1,7,36,75,100,60,16,1;
1,8,49,126,225,200,100,20,1; ...
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MAPLE
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T:=proc(n, k) if k<=n then binomial(n-floor(k/2), floor((k+1)/2))*binomial(n-floor((k+1)/2), floor(k/2)) else 0 fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form (from Emeric Deutsch)
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PROGRAM
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(PARI) T(n, k)=binomial(n-(k\2), (k+1)\2)*binomial(n-((k+1)\2), k\2)
(PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); polcoeff(polcoeff( 2/(1-X+X^2*Y^2-2*X*Y+sqrt((1-X+X^2*Y^2)^2-4*X^2*Y^2)), n, x), k, y)}
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CROSSREFS
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Cf. A104557, A091964.
Sequence in context: A124460 A144042 A122084 this_sequence A080853 A071922 A138028
Adjacent sequences: A104556 A104557 A104558 this_sequence A104560 A104561 A104562
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com) and Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 16 2005
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