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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

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. - R. J. Mathar, Nov 01 2007

EXAMPLE

Triangle starts:

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;

MAPLE

A008292 := proc(n, k) local j; add( (-1)^j*(k-j)^n*binomial(n+1, j), j=0..k) ; end: A049061 := proc(n, k) if k <= 0 or n <=0 or 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-Fran├žois Alcover, Jul 25 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A128613.

Sequence in context: A099927 A139332 A187617 * 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 February 24 06:13 EST 2018. Contains 299597 sequences. (Running on oeis4.)