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A362582
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Triangular array read by rows. T(n,k) is the number of alternating permutations of [2n+1] having exactly 2k elements to the left of 1, n >= 0, 0 <= k <= n.
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1
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1, 1, 1, 5, 6, 5, 61, 75, 75, 61, 1385, 1708, 1750, 1708, 1385, 50521, 62325, 64050, 64050, 62325, 50521, 2702765, 3334386, 3427875, 3438204, 3427875, 3334386, 2702765, 199360981, 245951615, 252857605, 253708455, 253708455, 252857605, 245951615, 199360981
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OFFSET
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0,4
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COMMENTS
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Here, w = w_1,w_2,...,w_(2n+1) is an alternating permutation if w_1 < w_2 > w_3 < ... < w_(2n) > w_(2n+1).
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LINKS
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FORMULA
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Sum_{n>=0} Sum_{k=0..n} T(n,k)*u^k*z^n/A000680(n) = 1/(E(-u*z)*E(-z)) where E(z) = Sum_{n>=0} z^n/A000680(n).
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EXAMPLE
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T(2,1) = 6 because we have: {2, 3, 1, 5, 4}, {2, 4, 1, 5, 3}, {2, 5, 1, 4, 3}, {3, 4, 1, 5, 2}, {3, 5, 1, 4, 2}, {4, 5, 1, 3, 2}.
Triangle begins
1;
1, 1;
5, 6, 5;
61, 75, 75, 61;
1385, 1708, 1750, 1708, 1385;
50521, 62325, 64050, 64050, 62325, 50521;
...
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MAPLE
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b:= proc(u, o) option remember; `if`(u+o=0, 1,
add(b(o-1+j, u-j), j=1..u))
end:
T:= (n, k)-> binomial(2*n, 2*k)*b(2*k, 0)*b(2*(n-k), 0):
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MATHEMATICA
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nn = 6; B[n_] := (2 n)!/2^n; e[z_] := Sum[z^n/B[n], {n, 0, nn}]; Map[Select[#, # > 0 &] &, Table[B[n], {n, 0, nn}] CoefficientList[Series[1/e[-u z]*1/e[-z], {z, 0, nn}], {z, u}]] // Grid
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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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