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Triangle read by rows: T(n,k) is the number of permutations of length n that have k same elements at the same positions with its inverse permutation for 0 <= k <= n.
2

%I #32 Oct 29 2024 12:55:08

%S 1,0,1,0,0,2,2,0,0,4,6,8,0,0,10,24,30,40,0,0,26,160,144,180,160,0,0,

%T 76,1140,1120,1008,840,700,0,0,232,8988,9120,8960,5376,4200,2912,0,0,

%U 764,80864,80892,82080,53760,30240,19656,12768,0,0,2620,809856,808640,808920,547200,336000,157248,95760,55680,0,0,9496

%N Triangle read by rows: T(n,k) is the number of permutations of length n that have k same elements at the same positions with its inverse permutation for 0 <= k <= n.

%H Alois P. Heinz, <a href="/A344901/b344901.txt">Rows n = 0..140, flattened</a>

%F T(n,k) = binomial(n,k)*A000085(k)*A038205(n-k).

%F From _Alois P. Heinz_, Oct 28 2024: (Start)

%F Sum_{k=0..n} k * T(n,k) = A052849(n) = A098558(n) for n>=2.

%F Sum_{k=0..n} (n-k) * T(n,k) = A052571(n).

%F Sum_{k=0..n} (-1)^k * T(n,k) = A000023(n).

%F T(n,0) + T(n,1) = A137482(n). (End)

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 0, 2;

%e 2, 0, 0, 4;

%e 6, 8, 0, 0, 10;

%e 24, 30, 40, 0, 0, 26;

%e 160, 144, 180, 160, 0, 0, 76;

%e 1140, 1120, 1008, 840, 700, 0, 0, 232;

%e 8988, 9120, 8960, 5376, 4200, 2912, 0, 0, 764;

%e ...

%p b:= proc(n, t) option remember; `if`(n=0, 1, add(b(n-j, t)*

%p binomial(n-1, j-1)*(j-1)!, j=`if`(t=1, 1..min(2, n), 3..n)))

%p end:

%p T:= (n, k)-> binomial(n, k)*b(k, 1)*b(n-k, 0):

%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Oct 28 2024

%Y Columns k=0-1 give: A038205, A221145.

%Y Row sums give A000142.

%Y Main diagonal gives A000085.

%Y Cf. A000023, A007318, A052571, A052849, A098558, A137482.

%K nonn,tabl

%O 0,6

%A _Mikhail Kurkov_, Jun 01 2021