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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.

%I A010029 #27 Jul 01 2022 06:17:46
%S A010029 1,1,1,3,3,1,12,11,11,56,53,3,87,321,309,53,693,2175,2119,11,680,5934,
%T A010029 17008,16687,309,8064,55674,150504,148329,53,5805,96370,572650,
%U A010029 1485465,1468457,2119,95575
%N 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.
%D A010029 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264, Table 7.6.1.
%H A010029 Vincenzo Librandi, <a href="/A010029/b010029.txt">Rows n = 1..50, flattened</a>
%F A010029 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 A010029 T(n,k) = A136123(n,[n/2]-k). - _R. J. Mathar_, Jul 01 2022
%e A010029 Triangle begins
%e A010029      1
%e A010029      1     1
%e A010029      3     3
%e A010029      1    12    11
%e A010029     11    56    53
%e A010029      3    87   321    309
%e A010029     53   693  2175   2119
%e A010029     11   680  5934  17008   16687
%e A010029    309  8064 55674 150504  148329
%e A010029     53  5805 96370 572650 1485465 1468457
%e A010029   2119 95575 ...
%e A010029   ...
%p A010029 A010029 := proc(n,k)
%p A010029     add( x^i*( ((1-y)*x-1)/((1-y)*x^2-1) )^i*i!,i=0..n+1) ;
%p A010029     coeftayl(%,x=0,n) ;
%p A010029     coeftayl(%,y=0,floor(n/2)-k) ;
%p A010029 end proc:
%p A010029 seq(seq( A010029(n,k),k=0..floor(n/2)),n=1..12) ; # _R. J. Mathar_, Jul 01 2022
%t A010029 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 A010029 Cf. A000255, A001277, A001278, A001279, A001280, A000142 (row sums), A136123 (rows reversed).
%K A010029 tabf,nonn,nice
%O A010029 1,4
%A A010029 _N. J. A. Sloane_