A161121 Triangle read by rows: T(n,k) is the number of fixed-point-free involutions of {1,2,...,2n} having k cycles with entries of the same parity (0 <= k <= 2*floor(n/2)).

1, 1, 2, 0, 1, 6, 0, 9, 24, 0, 72, 0, 9, 120, 0, 600, 0, 225, 720, 0, 5400, 0, 4050, 0, 225, 5040, 0, 52920, 0, 66150, 0, 11025, 40320, 0, 564480, 0, 1058400, 0, 352800, 0, 11025, 362880, 0, 6531840, 0, 17146080, 0, 9525600, 0, 893025, 3628800, 0, 81648000, 0

Offset: 0

Author

Emeric Deutsch, Jun 02 2009

Keywords

• nonn
• tabf

Comments

Row n contains 1 + 2*floor(n/2) terms.

Sum of row n = (2n-1)!! (A001147).

a(n,0) = n! (A000142).

a(2n,2n) = A001818(n).

Sum_{k>=0} k*T(n,k) = n*(n-1)*(2n-3)!! = A161122(n).

Examples

			T(3,2)=9 because we have (12)(35)(46), (14)(26)(35), (16)(24)(35), (23)(15)(46), (25)(13)(46), (34)(15)(26), (36)(15)(24), (45)(13)(26), (56)(13)(24).
Triangle starts:
    1;
    1;
    2,   0,   1;
    6,   0,   9;
   24,   0,  72,   0,   9;
  120,   0, 600,   0, 225;
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10081 (rows 0 to 141, flattened)

Crossrefs

Cf. A000142, A001147, A001818, A161119, A161120, A161122.

Programs

Maple

T := proc (n, k) if n < k then 0 elif `mod`(k, 2) = 0 then binomial(n, k)^2*factorial(n-k)*(product(2*j-1, j = 1 .. (1/2)*k))^2 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. 2*floor((1/2)*n)) end do; # yields sequence in triangular form

Mathematica

T[n_, k_] := If[EvenQ[k], (n-k)! Binomial[n, k]^2 ((k-1)!!)^2, 0];
Table[T[n, k], {n, 0, 10}, {k, 0, 2 Quotient[n, 2]}] // Flatten (* Jean-François Alcover, Feb 01 2023 *)

Formula

T(n,k) = (n-k)!*binomial(n,k)^2*((k-1)!!)^2 if k is even; T(n,k) = 0 if k is odd.