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
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
- Sami H. Assaf, Cyclic derangements, arXiv:1002.3138 [math.CO], 2010.
- Hilarion L. M. Faliharimalala and Jiang Zeng, Derangements and Euler's difference table for C_l wr S_n, Electron. J. Combin. 15 (2008), #R65.
- Wikipedia, Generalized symmetric group.
Crossrefs
Programs
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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
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).
Comments