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Search: id:A128612
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| A128612 |
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Triangle T(n,k) read by rows: number of permutations in [n] with exactly k ascents that have an even number of inversions. |
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+0
1
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| 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; listen)
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OFFSET
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1,5
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LINKS
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S. Tanimoto, A new approach to signed Eulerian numbers
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FORMULA
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T(n,k)=(1/2) * [A008292(n,n-k)+A049061(n,n-k)], n>=1, 0<=k<n. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 01 2007
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EXAMPLE
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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;
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MAPLE
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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 (mathar(AT)strw.leidenuniv.nl), Nov 01 2007
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CROSSREFS
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Sequence in context: A008518 A139332 A099927 this_sequence A060854 A091378 A156045
Adjacent sequences: A128609 A128610 A128611 this_sequence A128613 A128614 A128615
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KEYWORD
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nonn,tabl
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AUTHOR
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Ralf Stephan, May 08 2007
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EXTENSIONS
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More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 01 2007
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