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 A306234 Number T(n,k) of occurrences of k in a (signed) displacement set of a permutation of [n] divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows. 18
 1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 5, 13, 15, 13, 5, 1, 1, 7, 28, 67, 76, 67, 28, 7, 1, 1, 9, 49, 179, 411, 455, 411, 179, 49, 9, 1, 1, 11, 76, 375, 1306, 2921, 3186, 2921, 1306, 375, 76, 11, 1, 1, 13, 109, 679, 3181, 10757, 23633, 25487, 23633, 10757, 3181, 679, 109, 13, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 LINKS Alois P. Heinz, Rows n = 1..142, flattened Wikipedia, Permutation FORMULA T(n,k) = T(n,-k). T(n,k) = -1/|k|! * Sum_{j=1..n} (-1)^j * binomial(n-|k|,j) * (n-j)!. T(n,k) = (n-|k|)! [x^(n-|k|)] (1-exp(-x))/(1-x)^(|k|+1). T(n+1,n) = 1. T(n,k) = A306461(n,k) / |k|!. Sum_{k=1-n..n-1} |k|! * T(n,k) = A306455(n). EXAMPLE Triangle T(n,k) begins:   :                                 1                              ;   :                           1,    1,    1                        ;   :                     1,    3,    4,    3,    1                  ;   :               1,    5,   13,   15,   13,    5,   1             ;   :          1,   7,   28,   67,   76,   67,   28,   7,  1         ;   :      1,  9,  49,  179,  411,  455,  411,  179,  49,  9,  1     ;   :  1, 11, 76, 375, 1306, 2921, 3186, 2921, 1306, 375, 76, 11, 1  ; MAPLE b:= proc(s, d) option remember; (n-> `if`(n=0, add(x^j, j=d),       add(b(s minus {i}, d union {n-i}), i=s)))(nops(s))     end: T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({\$1..n}, {})): seq(T(n), n=1..8); # second Maple program: T:= (n, k)-> -add((-1)^j*binomial(n-abs(k), j)*(n-j)!, j=1..n)/abs(k)!: seq(seq(T(n, k), k=1-n..n-1), n=1..9); MATHEMATICA T[n_, k_] := (-1/Abs[k]!) Sum[(-1)^j Binomial[n-Abs[k], j] (n-j)!, {j, 1, n}]; Table[T[n, k], {n, 1, 9}, {k, 1-n, n-1}] // Flatten (* Jean-François Alcover, Feb 15 2021 *) CROSSREFS Columns k=0-10 give (offsets may differ): A002467, A180191, A324352, A324353, A324354, A324355, A324356, A324357, A324358, A324359, A324360. Row sums give A306525. T(n+1,n) gives A000012. T(n+2,n) gives A005408. T(n+2,n-1) gives A056107. T(2n,n) gives A324361. Cf. A000142, A306455, A306461, A324224, A324362. Sequence in context: A111028 A201162 A096646 * A290057 A249790 A302713 Adjacent sequences:  A306231 A306232 A306233 * A306235 A306236 A306237 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, Feb 17 2019 STATUS approved

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Last modified June 21 04:10 EDT 2021. Contains 345354 sequences. (Running on oeis4.)