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 A245405 Number A(n,k) of endofunctions on [n] such that no element has a preimage of cardinality k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 14
 1, 1, 1, 1, 0, 2, 1, 1, 2, 6, 1, 1, 2, 3, 24, 1, 1, 4, 9, 40, 120, 1, 1, 4, 24, 76, 205, 720, 1, 1, 4, 27, 208, 825, 2556, 5040, 1, 1, 4, 27, 252, 2325, 10206, 24409, 40320, 1, 1, 4, 27, 256, 3025, 31956, 143521, 347712, 362880, 1, 1, 4, 27, 256, 3120, 44406, 520723, 2313200, 4794633, 3628800 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened FORMULA A(n,k) = n! * [x^n] (exp(x)-x^k/k!)^n. EXAMPLE Square array A(n,k) begins: 0 :   1,    1,     1,     1,     1,     1,     1, ... 1 :   1,    0,     1,     1,     1,     1,     1, ... 2 :   2,    2,     2,     4,     4,     4,     4, ... 3 :   6,    3,     9,    24,    27,    27,    27, ... 4 :  24,   40,    76,   208,   252,   256,   256, ... 5 : 120,  205,   825,  2325,  3025,  3120,  3125, ... 6 : 720, 2556, 10206, 31956, 44406, 46476, 46650, ... MAPLE b:= proc(n, i, k) option remember; `if`(n=0 and i=0, 1,       `if`(i<1, 0, add(`if`(j=k, 0, b(n-j, i-1, k)*        binomial(n, j)), j=0..n)))     end: A:= (n, k)-> b(n\$2, k): seq(seq(A(n, d-n), n=0..d), d=0..14); MATHEMATICA nn = n; f[m_]:=Flatten[Table[m[[j, i - j + 1]], {i, 1, Length[m]}, {j, 1, i}]]; f[Transpose[Table[Prepend[Table[n! Coefficient[Series[(Exp[x] -x^k/k!)^n, {x, 0, nn}], x^n], {n, 1, 10}], 1], {k, 0, 10}]]] (* Geoffrey Critzer, Jan 31 2015 *) CROSSREFS Column k=0-10 give: A000142, A231797, A245406, A245407, A245408, A245409, A245410, A245411, A245412, A245413, A245414. Main diagonal gives A061190. A(n,n+1) gives A000312. Sequence in context: A163982 A246661 A246660 * A233543 A156588 A278543 Adjacent sequences:  A245402 A245403 A245404 * A245406 A245407 A245408 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jul 21 2014 STATUS approved

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Last modified July 28 19:33 EDT 2021. Contains 346335 sequences. (Running on oeis4.)