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 A161123 Triangle read by rows: T(n,k) is the number of fixed-point-free involutions of {1,2,...,2n} having k inversions (0<=k<=n(2n-1)). 1
 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 3, 0, 6, 0, 9, 0, 12, 0, 14, 0, 15, 0, 14, 0, 12, 0, 9, 0, 6, 0, 3, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 10, 0, 19, 0, 31, 0, 45, 0, 60, 0, 74, 0, 86, 0, 94, 0, 97, 0, 94, 0, 86, 0, 74, 0, 60, 0, 45, 0, 31, 0, 19 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,16 COMMENTS Sum of entries in row n is (2n-1)!! =A001147(n). Row n has 1+2n(n-1) entries. Sum(k*T(n,k), k>=0) = (2n-1)!!*n^2 = A161124(n). A128080 is the same triangle with the 0's deleted. LINKS FORMULA Generating polynomial of row n is P_n(q)=[q/(1-q^2)]^n*Product(1-q^{4j-2}, j=1..n). EXAMPLE T(3,11)=3 because we have 465132, 546213, and 632541. Triangle starts: 1; 0,1; 0,0,1,0,1,0,1; 0,0,0,1,0,2,0,3,0,3,0,3,0,2,0,1. MAPLE f := proc (n) options operator, arrow: q^n*(product(1-q^(4*j-2), j = 1 .. n))/(1-q^2)^n end proc: for n from 0 to 4 do P[n] := sort(expand(simplify(f(n)))) end do: for n from 0 to 4 do seq(coeff(P[n], q, j), j = 0 .. n*(2*n-1)) end do; # yields sequence in triangular form CROSSREFS Sequence in context: A011374 A189230 A214000 * A035442 A213177 A035376 Adjacent sequences:  A161120 A161121 A161122 * A161124 A161125 A161126 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Jun 05 2009 STATUS approved

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