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A247469
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Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k weak peaks.
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1
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1, 1, 2, 3, 1, 5, 2, 1, 8, 5, 3, 1, 13, 11, 8, 4, 1, 21, 24, 19, 12, 5, 1, 34, 52, 44, 31, 17, 6, 1, 55, 112, 101, 76, 48, 23, 7, 1, 89, 241, 230, 183, 125, 71, 30, 8, 1, 144, 518, 524, 436, 315, 197, 101, 38, 9, 1, 233, 1113, 1193, 1037, 781, 520, 299, 139, 47, 10, 1
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OFFSET
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0,3
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COMMENTS
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B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps.
A weak peak in a lattice path is a vertex on the top of a hump. A hump is an upstep followed by 0 or more flatsteps followed by a downstep. For example, the weighted lattice path Hu*duu*h*H*dd has 4 weak peaks (shown by the stars).
Row n contains n-1 entries (n>=2).
Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
T(n,0) = A000045(n+1) (the Fibonacci numbers).
Sum(k*T(n,k), k=0..n) = A247470(n).
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LINKS
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FORMULA
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G.f. G = G(t,z) satisfies G = 1 + z*G + z^2*G + z^3*G*(G - 1/(1 - z - z^2) + t/(1 - t*z - t*z^2)).
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EXAMPLE
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Row 3 is 3, 1 because B(3) = {u*d, hH, Hh, hhh} (weak peaks shown by *).
Triangle starts:
1;
1;
2;
3,1;
5,2,1;
8,5,3,1;
13,11,8,4,1;
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MAPLE
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eq := G = 1+z*G+z^2*G+z^3*G*(G-1/(1-z-z^2)+t/(1-t*z-t*z^2)): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 17 do seq(coeff(P[n], t, k), k = 0 .. n-2) end do; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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