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A096793
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Triangle read by rows: a(n,k) is the number of Dyck n-paths containing k odd-length ascents.
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1
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1, 0, 1, 1, 0, 1, 0, 4, 0, 1, 3, 0, 10, 0, 1, 0, 21, 0, 20, 0, 1, 12, 0, 84, 0, 35, 0, 1, 0, 120, 0, 252, 0, 56, 0, 1, 55, 0, 660, 0, 630, 0, 84, 0, 1, 0, 715, 0, 2640, 0, 1386, 0, 120, 0, 1, 273, 0, 5005, 0, 8580, 0, 2772, 0, 165, 0, 1, 0, 4368, 0, 25025, 0, 24024, 0, 5148, 0, 220, 0, 1
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OFFSET
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0,8
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COMMENTS
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a(n,k)=0 unless k and n have the same parity and 0 <= k <= n.
Sum_{k=0..n} k*a(n,k) = A014300(n).
For the case of even-length ascents see A143950. (End)
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LINKS
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FORMULA
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a(n, k) = binomial((n+k)/2, (n-k)/2)*binomial((3n-k)/2+1, (n+k)/2)/((3n-k)/2+1).
Equivalently, a(2n+k, k) = binomial(3n+k, k)*T(n) where T(n) = binomial(3n, n)/(2n+1) is A001764. Proof: Given a Dyck (2n+k)-path with k ascents of odd length, delete the peaks (UD) that terminate odd-length ascents. This is a mapping to Dyck (2n)-paths all of whose ascents have even length; there are T[n] such paths. The mapping is clearly onto and is binomial(3n+k, k)-to-1 as follows. A Dyck (2n)-path all of whose ascents have even length has exactly 3n+1 vertices that are (i) not incident with an upstep, or (ii) incident with an upstep and at even distance (possibly 0) from the start of the ascent they lie in. The k deleted UDs can be inserted arbitrarily at these vertices, repetition allowed, to get the preimages -- binomial(3n+k, k) choices.
G.f.: G(z, t) + H(z, t) where G satisfies G^3*(t^2 - 1)*z^2 - G^2*t*z*(2 + t*z) + G*(1 + 2*t*z) - 1 = 0 and H satisfies H^3*(t^2 - 1)*z^2 + H^2*t*z*(2 + t*z) - H*t^2*(1 - t*z) + t^3*z = 0. Here z marks size (n) and t marks number of odd-length ascents (k). G is gf for paths that start with an even-length ascent and H is gf for paths that start with an odd-length ascent. - David Callan, Sep 03 2005
G.f. G=G(t,z) satisfies G = 1 + zG(t + zG)/(1 - z^2*G^2).
The trivariate g.f. H=H(t,s,z), where t(s) marks odd-length (even-length) ascents satisfies H = 1 + zH(t+szH)/(1-z^2*H^2). (End)
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EXAMPLE
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Table begins
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n |k = 0 1 2 3 4 5 6 7 8
--+---------------------------------------------
0 | 1
1 | 0, 1
2 | 1, 0, 1
3 | 0, 4, 0, 1
4 | 3, 0, 10, 0, 1
5 | 0, 21, 0, 20, 0, 1
6 | 12, 0, 84, 0, 35, 0, 1
7 | 0, 120, 0, 252, 0, 56, 0, 1
8 | 55, 0, 660, 0, 630, 0, 84, 0, 1
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a(4,0)=3 because the Dyck 4-paths containing no odd-length ascents are UUUUDDDD,UUDUUDDD,UUDDUUDD.
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MATHEMATICA
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bi[n_, k_] := If[IntegerQ[k], Binomial[n, k], 0]; TableForm[Table[bi[(n+k)/2, (n-k)/2]bi[(3n-k)/2+1, (n+k)/2]/((3n-k)/2+1), {n, 0, 10}, {k, 0, n}]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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