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A245687 Number T(n,k) of endofunctions on [n] such that the minimal cardinality of the nonempty preimages equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 4
1, 0, 1, 0, 2, 2, 0, 24, 0, 3, 0, 216, 36, 0, 4, 0, 2920, 200, 0, 0, 5, 0, 44100, 2250, 300, 0, 0, 6, 0, 799134, 22932, 1470, 0, 0, 0, 7, 0, 16429504, 342608, 3136, 1960, 0, 0, 0, 8, 0, 382625856, 4638384, 147168, 9072, 0, 0, 0, 0, 9, 0, 9918836100, 79610850, 1522800, 18900, 11340, 0, 0, 0, 0, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

T(0,0) = 1 by convention.

LINKS

Alois P. Heinz, Rows n = 0..100, flattened

EXAMPLE

Triangle T(n,k) begins:

  1;

  0,        1;

  0,        2,      2;

  0,       24,      0,    3;

  0,      216,     36,    0,    4;

  0,     2920,    200,    0,    0,  5;

  0,    44100,   2250,  300,    0,  0,  6;

  0,   799134,  22932, 1470,    0,  0,  0,  7;

  0, 16429504, 342608, 3136, 1960,  0,  0,  0,  8;

MAPLE

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

      b(n, i-1, k) +add(b(n-j, i-1, k)/j!, j=k..n)))

    end:

T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), `if`(k=n, n,

    `if`(k>=(n+1)/2, 0, n!*(b(n$2, k)-b(n$2, k+1))))):

seq(seq(T(n, k), k=0..n), n=0..12);

MATHEMATICA

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + Sum[b[n-j, i-1, k]/j!, {j, k, n}]]]; T[n_, k_] := If[k == 0, If[n == 0, 1, 0], If[k == n, n, If[k >= (n+1)/2, 0, n!*(b[n, n, k] - b[n, n, k+1])]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Feb 02 2015, after Alois P. Heinz *)

CROSSREFS

T(n,1) = n*A241581(n) for n>0.

Rows sums give A000312.

Main diagonal gives A028310.

T(2n,n) gives A273325.

Cf. A019575 (the same for maximal cardinality).

Sequence in context: A151339 A228273 A069521 * A228617 A119836 A158112

Adjacent sequences:  A245684 A245685 A245686 * A245688 A245689 A245690

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 29 2014

STATUS

approved

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Last modified February 17 12:32 EST 2020. Contains 331996 sequences. (Running on oeis4.)