A010029 Irregular triangle read by rows: T(n,k) (n>=1, 0 <= k <= floor(n/2)) = number of permutations of 1..n with exactly floor(n/2) - k runs of consecutive pairs up.

1, 1, 1, 3, 3, 1, 12, 11, 11, 56, 53, 3, 87, 321, 309, 53, 693, 2175, 2119, 11, 680, 5934, 17008, 16687, 309, 8064, 55674, 150504, 148329, 53, 5805, 96370, 572650, 1485465, 1468457, 2119, 95575

Offset

1,4

References

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264, Table 7.6.1.

Links

Vincenzo Librandi, Rows n = 1..50, flattened

Formula

G.f.: Sum_{n>=0} n! *( ((1-y)*x^2-x)/((1-y)*x^2-1) )^n, for the triangle read right-to-left. - Vladeta Jovovic, Nov 21 2007

T(n,k) = A136123(n,[n/2]-k). - R. J. Mathar, Jul 01 2022

Example

Triangle begins
     1
     1     1
     3     3
     1    12    11
    11    56    53
     3    87   321    309
    53   693  2175   2119
    11   680  5934  17008   16687
   309  8064 55674 150504  148329
    53  5805 96370 572650 1485465 1468457
  2119 95575 ...
  ...

Maple

A010029 := proc(n, k)
    add( x^i*( ((1-y)*x-1)/((1-y)*x^2-1) )^i*i!, i=0..n+1) ;
    coeftayl(%, x=0, n) ;
    coeftayl(%, y=0, floor(n/2)-k) ;
end proc:
seq(seq( A010029(n, k), k=0..floor(n/2)), n=1..12) ; # R. J. Mathar, Jul 01 2022

Mathematica

max = 16; coes = CoefficientList[ Series[ Sum[ n!*(((1 - y)*x^2 - x)/((1 - y)*x^2 - 1))^n, {n, 0, max}], {x, 0, max}, {y, 0, max}], {x, y}]; Table[ Table[ coes[[n, k]] , {k, 1, Floor[(n + 1)/2]}] // Reverse, {n, 2, max - 4}] // Flatten (* Jean-François Alcover, Jan 10 2013, after Vladeta Jovovic *)

Crossrefs

Cf. A000255, A001277, A001278, A001279, A001280, A000142 (row sums), A136123 (rows reversed).

Sequence in context: A185418 A050609 A120870 * A143603 A094021 A062746

Adjacent sequences: A010026 A010027 A010028 * A010030 A010031 A010032

Keyword

tabf,nonn,nice

Author

N. J. A. Sloane

Status

approved