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A094718
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Array T read by antidiagonals: T(n,k) is the number of involutions avoiding 132 and 12...k.
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14
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0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 4, 1, 0, 1, 2, 3, 5, 4, 1, 0, 1, 2, 3, 6, 8, 8, 1, 0, 1, 2, 3, 6, 9, 13, 8, 1, 0, 1, 2, 3, 6, 10, 18, 21, 16, 1, 0, 1, 2, 3, 6, 10, 19, 27, 34, 16, 1, 0, 1, 2, 3, 6, 10, 20, 33, 54, 55, 32, 1, 0, 1, 2, 3, 6, 10, 20, 34, 61, 81, 89, 32, 1
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OFFSET
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1,8
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COMMENTS
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Also, number of paths along a corridor with width k, starting from one side (from H. Bottomley's comment in A061551).
Rows converge to binomial(n,floor(n/2)) (A001405).
Note that the rows and columns start at 1, which for example obscures the fact that the first row refers to A000007 and not to A000004. A better choice is the indexing 0 <= k and 0 <= n. The Maple program below uses this indexing and builds only on the roots of -1. - Peter Luschny, Sep 17 2020
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LINKS
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Robert Dutton Fray and David Paul Roselle, Weighted lattice paths, Pacific Journal of Mathematics, 37(1) (1971), 85-96.
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FORMULA
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G.f. for k-th row: 1/(x*U(k, 1/(2*x))) * Sum_{j=0..k-1} U(j, 1/(2*x)), with U(k, x) the Chebyshev polynomials of second kind. [Clarified by Jean-François Alcover, Nov 17 2018]
T(n, k) = (1/(n+1))*Sum_{j=1..n, j odd} (2 + [j, n]) * [j, n]^k where [j, n] := (-1)^(j/(n+1)) - (-1)^((n-j+1)/(n+1)). - Peter Luschny, Sep 17 2020
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EXAMPLE
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Array begins
0 0 0 0 0 0 0 0 0 0 ...
1 1 1 1 1 1 1 1 1 1 ...
1 2 2 4 4 8 8 16 16 32 ...
1 2 3 5 8 13 21 34 55 89 ...
1 2 3 6 9 18 27 54 81 162 ...
1 2 3 6 10 19 33 61 108 197 ...
1 2 3 6 10 20 34 68 116 232 ...
1 2 3 6 10 20 35 69 124 241 ...
1 2 3 6 10 20 35 70 125 250 ...
1 2 3 6 10 20 35 70 126 251 ...
...
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MAPLE
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X := (j, n) -> (-1)^(j/(n+1)) - (-1)^((n-j+1)/(n+1)):
R := n -> select(k -> type(k, odd), [$(1..n)]):
T := (n, k) -> add((2 + X(j, n))*X(j, n)^k, j in R(n))/(n+1):
seq(lprint([n], seq(simplify(T(n, k)), k=0..10)), n=0..9); # Peter Luschny, Sep 17 2020
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MATHEMATICA
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U = ChebyshevU;
m = maxExponent = 14;
row[1] = Array[0&, m];
row[k_] := 1/(x U[k, 1/(2x)])*Sum[U[j, 1/(2x)], {j, 0, k-1}] + O[x]^m // CoefficientList[#, x]& // Rest;
T = Table[row[n], {n, 1, m}];
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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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