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Search: id:A114690
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| A114690 |
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Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k weak ascents (1<=k<=ceil(n/2)). A Motzkin path of length n is a lattice path from (0,0) to (n,0) consisting of U=(1,1), D=(1,-1), and H=(1,0) steps and never going below the x-axis. A weak ascent in a Motzkin path is a maximal sequence of consecutive U and H steps. |
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1
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| 1, 2, 3, 1, 5, 4, 8, 12, 1, 13, 31, 7, 21, 73, 32, 1, 34, 162, 116, 11, 55, 344, 365, 70, 1, 89, 707, 1041, 335, 16, 144, 1416, 2762, 1340, 135, 1, 233, 2778, 6932, 4726, 820, 22, 377, 5358, 16646, 15176, 4039, 238, 1, 610, 10188, 38560, 45305, 17157, 1785, 29, 987
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OFFSET
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1,2
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COMMENT
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Row n has ceil(n/2) terms. Row sums are the Motzkin numbers (A001006). Column 1 yield the Fibonacci numbers (A000045). Sum(k*T(n,k)=A005773(n).
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FORMULA
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G.f. G=G(t, z) satisfies G=z(t+G)(1+z+zG).
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EXAMPLE
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T(4,2)=4 because we have (HU)D(H),(U)D(HH),(U)D(U)D, and (UH)D(H) (the weak ascents are shown between parentheses).
Triangle starts:
1;
2;
3,1;
5,4;
8,12,1;
13,31,7;
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MAPLE
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G:=(1-t*z^2-z-z^2-sqrt(1-2*t*z^2-2*z-z^2+t^2*z^4-2*t*z^3-2*z^4*t+2*z^3+z^4))/2/z^2: Gser:=simplify(series(G, z=0, 18)): for n from 1 to 15 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 15 do seq(coeff(P[n], t^j), j=1..ceil(n/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001006, A005773, A000045, A114655.
Sequence in context: A114711 A002472 A060116 this_sequence A049274 A130508 A055231
Adjacent sequences: A114687 A114688 A114689 this_sequence A114691 A114692 A114693
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 24 2005
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