A010029 - OEIS

%I #27 Jul 01 2022 06:17:46

%S 1,1,1,3,3,1,12,11,11,56,53,3,87,321,309,53,693,2175,2119,11,680,5934,

%T 17008,16687,309,8064,55674,150504,148329,53,5805,96370,572650,

%U 1485465,1468457,2119,95575

%N 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.

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

%H Vincenzo Librandi, <a href="/A010029/b010029.txt">Rows n = 1..50, flattened</a>

%F 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

%F T(n,k) = A136123(n,[n/2]-k). - _R. J. Mathar_, Jul 01 2022

%e Triangle begins

%e      1

%e      1     1

%e      3     3

%e      1    12    11

%e     11    56    53

%e      3    87   321    309

%e     53   693  2175   2119

%e     11   680  5934  17008   16687

%e    309  8064 55674 150504  148329

%e     53  5805 96370 572650 1485465 1468457

%e   2119 95575 ...

%e   ...

%p A010029 := proc(n,k)

%p     add( x^i*( ((1-y)*x-1)/((1-y)*x^2-1) )^i*i!,i=0..n+1) ;

%p     coeftayl(%,x=0,n) ;

%p     coeftayl(%,y=0,floor(n/2)-k) ;

%p end proc:

%p seq(seq( A010029(n,k),k=0..floor(n/2)),n=1..12) ; # _R. J. Mathar_, Jul 01 2022

%t 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_ *)

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

%K tabf,nonn,nice

%O 1,4

%A _N. J. A. Sloane_