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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. 3
 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 (list; graph; refs; listen; history; text; internal format)
 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,changed AUTHOR STATUS approved

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Last modified July 2 07:50 EDT 2022. Contains 354985 sequences. (Running on oeis4.)