A320032 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the e.g.f. exp(-x)/(1 - k*x).

1, 1, -1, 1, 0, 1, 1, 1, 1, -1, 1, 2, 5, 2, 1, 1, 3, 13, 29, 9, -1, 1, 4, 25, 116, 233, 44, 1, 1, 5, 41, 299, 1393, 2329, 265, -1, 1, 6, 61, 614, 4785, 20894, 27949, 1854, 1, 1, 7, 85, 1097, 12281, 95699, 376093, 391285, 14833, -1, 1, 8, 113, 1784, 26329, 307024, 2296777, 7897952, 6260561, 133496, 1

Offset

0,12

Comments

For n > 0 and k > 0, A(n,k) gives the number of derangements of the generalized symmetric group S(k,n), which is the wreath product of Z_k by S_n. - Peter Kagey, Apr 07 2020

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Sami H. Assaf, Cyclic derangements, arXiv:1002.3138 [math.CO], 2010.

Wikipedia, Generalized symmetric group.

Formula

E.g.f. of column k: exp(-x)/(1 - k*x).

A(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j!*k^j.

A(n,k) = (-1)^n*2F0(1,-n; ; k).

Example

E.g.f. of column k: A_k(x) = 1 + (k - 1)*x/1! + (2*k^2 - 2*k + 1)*x^2/2! + (6*k^3 - 6*k^2 + 3*k - 1)*x^3/3! + (24*k^4 - 24*k^3 + 12*k^2 - 4*k + 1)*x^4/4! + ...
Square array begins:
   1,   1,     1,      1,      1,       1,  ...
  -1,   0,     1,      2,      3,       4,  ...
   1,   1,     5,     13,     25,      41,  ...
  -1,   2,    29,    116,    299,     614,  ...
   1,   9,   233,   1393,   4785,   12281,  ...
  -1,  44,  2329,  20894,  95699,  307024,  ...

Maple

A:= proc(n, k) option remember;
     `if`(n=0, 1, k*n*A(n-1, k)+(-1)^n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 07 2020

Mathematica

Table[Function[k, n! SeriesCoefficient[Exp[-x]/(1 - k x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, (-1)^n HypergeometricPFQ[{1, -n}, {}, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Crossrefs

Columns k=0..5 give A033999, A000166, A000354, A000180, A001907, A001908.

Main diagonal gives A319392.

Cf. A320031.

Sequence in context: A371751 A196527 A064334 * A270061 A342059 A061176

Adjacent sequences: A320029 A320030 A320031 * A320033 A320034 A320035

Keyword

sign,tabl

Author

Ilya Gutkovskiy, Oct 03 2018

Status

approved