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A334715 A(n,k) = !n + [n > 0] * (k * n!), where !n = A000166(n) is subfactorial of n and [] is an Iverson bracket; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A334715 #22 Feb 11 2021 10:36:29
%S A334715 1,1,0,1,1,1,1,2,3,2,1,3,5,8,9,1,4,7,14,33,44,1,5,9,20,57,164,265,1,6,
%T A334715 11,26,81,284,985,1854,1,7,13,32,105,404,1705,6894,14833,1,8,15,38,
%U A334715 129,524,2425,11934,55153,133496,1,9,17,44,153,644,3145,16974,95473,496376,1334961
%N A334715 A(n,k) = !n + [n > 0] * (k * n!), where !n = A000166(n) is subfactorial of n and [] is an Iverson bracket; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A334715 Alois P. Heinz, <a href="/A334715/b334715.txt">Antidiagonals n = 0..140, flattened</a>
%H A334715 Wikipedia, <a href="https://en.wikipedia.org/wiki/Derangement">Derangement</a>
%H A334715 Wikipedia, <a href="https://en.wikipedia.org/wiki/Iverson_bracket">Iverson bracket</a>
%F A334715 E.g.f. of column k: (k*exp(x)*x+1)*exp(-x)/(1-x).
%F A334715 A(n,k) = A000166(n) + [n > 0] * (k * n!).
%F A334715 A(n,k) = (k-1)*n + 1 if n<2, A(n,k) = n*A(n-1, k) + (-1)^n if n>=2.
%e A334715 Square array A(n,k) begins:
%e A334715      1,    1,     1,     1,     1,     1,     1,     1, ...
%e A334715      0,    1,     2,     3,     4,     5,     6,     7, ...
%e A334715      1,    3,     5,     7,     9,    11,    13,    15, ...
%e A334715      2,    8,    14,    20,    26,    32,    38,    44, ...
%e A334715      9,   33,    57,    81,   105,   129,   153,   177, ...
%e A334715     44,  164,   284,   404,   524,   644,   764,   884, ...
%e A334715    265,  985,  1705,  2425,  3145,  3865,  4585,  5305, ...
%e A334715   1854, 6894, 11934, 16974, 22014, 27054, 32094, 37134, ...
%e A334715   ...
%p A334715 A:= proc(n, k) option remember; `if`(n<2,
%p A334715       (k-1)*n+1, n*A(n-1, k)+(-1)^n)
%p A334715     end:
%p A334715 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A334715 A[n_, k_] := Subfactorial[n] + Boole[n>0] k n!;
%t A334715 Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 11 2021 *)
%Y A334715 Columns k=0-3 give: A000166, A001120, A110043, A110149.
%Y A334715 Rows n=0-3 give: A000012, A001477, A005408, A016933.
%Y A334715 Main diagonal gives A334716.
%Y A334715 Cf. A000142.
%K A334715 nonn,tabl
%O A334715 0,8
%A A334715 _Alois P. Heinz_, May 08 2020