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 A128612 Triangle T(n,k) read by rows: number of permutations in [n] with exactly k ascents that have an even number of inversions. 2
 1, 0, 1, 0, 2, 1, 1, 5, 5, 1, 1, 14, 30, 14, 1, 0, 28, 155, 147, 29, 1, 0, 56, 605, 1208, 586, 64, 1, 1, 127, 2133, 7819, 7819, 2133, 127, 1, 1, 262, 7288, 44074, 78190, 44074, 7288, 262, 1, 0, 496, 23947, 227623, 655039, 655315, 227569, 23893, 517, 1, 0, 992, 76305, 1102068, 4868556, 7862124, 4869558, 1101420, 76332, 1044, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 LINKS Alois P. Heinz, Rows n = 1..141, flattened Jason Fulman, Gene B. Kim, Sangchul Lee, T. Kyle Petersen, On the joint distribution of descents and signs of permutations, arXiv:1910.04258 [math.CO], 2019. S. Tanimoto, A new approach to signed Eulerian numbers, arXiv:math/0602263 [math.CO], 2006. FORMULA T(n,k) = (1/2) * [A008292(n,n-k)+A049061(n,n-k)], n>=1, 0<=k n then 0; elif n = 1 then 1 ; elif n mod 2 = 0 then A049061(n-1, k)-A049061(n-1, k-1) ; else k*A049061(n-1, k)+(n-k+1)*A049061(n-1, k-1) ; fi ; end: A128612 := proc(n, k) (A008292(n, n-k)+A049061(n, n-k))/2 ; end: for n from 1 to 11 do for k from 0 to n-1 do printf("%d, ", A128612(n, k)) ; od: od: # R. J. Mathar, Nov 01 2007 # second Maple program: b:= proc(u, o, i) option remember; expand(`if`(u+o=0, 1-i,        add(b(u+j-1, o-j, irem(i+u+j-1, 2)), j=1..o)*x+        add(b(u-j, o+j-1, irem(i+u-j, 2)), j=1..u)))     end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n, 0\$2)): seq(T(n), n=1..14);  # Alois P. Heinz, May 02 2017 MATHEMATICA b[u_, o_, i_] := b[u, o, i] = Expand[If[u + o == 0, 1 - i, Sum[b[u + j - 1, o - j, Mod[i + u + j - 1, 2]], {j, 1, o}]*x + Sum[b[u - j, o + j - 1, Mod[i + u - j, 2]], {j, 1, u}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[n, 0, 0]]; Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jul 25 2017, after Alois P. Heinz *) CROSSREFS Cf. A008292, A049061, A128613. Sequence in context: A139332 A187617 A306344 * A284731 A211400 A060854 Adjacent sequences:  A128609 A128610 A128611 * A128613 A128614 A128615 KEYWORD nonn,tabl AUTHOR Ralf Stephan, May 08 2007 EXTENSIONS More terms from R. J. Mathar, Nov 01 2007 STATUS approved

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Last modified January 22 16:37 EST 2020. Contains 331152 sequences. (Running on oeis4.)