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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. 6
 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, 11, 26, 81, 284, 985, 1854, 1, 7, 13, 32, 105, 404, 1705, 6894, 14833, 1, 8, 15, 38, 129, 524, 2425, 11934, 55153, 133496, 1, 9, 17, 44, 153, 644, 3145, 16974, 95473, 496376, 1334961 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened Wikipedia, Derangement Wikipedia, Iverson bracket FORMULA E.g.f. of column k: (k*exp(x)*x+1)*exp(-x)/(1-x). A(n,k) = A000166(n) + [n > 0] * (k * n!). A(n,k) = (k-1)*n + 1 if n<2, A(n,k) = n*A(n-1, k) + (-1)^n if n>=2. EXAMPLE Square array A(n,k) begins:      1,    1,     1,     1,     1,     1,     1,     1, ...      0,    1,     2,     3,     4,     5,     6,     7, ...      1,    3,     5,     7,     9,    11,    13,    15, ...      2,    8,    14,    20,    26,    32,    38,    44, ...      9,   33,    57,    81,   105,   129,   153,   177, ...     44,  164,   284,   404,   524,   644,   764,   884, ...    265,  985,  1705,  2425,  3145,  3865,  4585,  5305, ...   1854, 6894, 11934, 16974, 22014, 27054, 32094, 37134, ...   ... MAPLE A:= proc(n, k) option remember; `if`(n<2,       (k-1)*n+1, n*A(n-1, k)+(-1)^n)     end: seq(seq(A(n, d-n), n=0..d), d=0..10); MATHEMATICA A[n_, k_] := Subfactorial[n] + Boole[n>0] k n!; Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *) CROSSREFS Columns k=0-3 give: A000166, A001120, A110043, A110149. Rows n=0-3 give: A000012, A001477, A005408, A016933. Main diagonal gives A334716. Cf. A000142. Sequence in context: A107357 A251718 A182457 * A026105 A303868 A060475 Adjacent sequences:  A334712 A334713 A334714 * A334716 A334717 A334718 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, May 08 2020 STATUS approved

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Last modified January 22 10:13 EST 2022. Contains 350481 sequences. (Running on oeis4.)