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 A245733 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality k exists and, if j is the largest value with a nonempty preimage, the preimage cardinality of i is >=k for all i<=j and equal to k for at least one i<=j; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 13
 1, 0, 1, 1, 2, 1, 14, 12, 0, 1, 181, 68, 6, 0, 1, 2584, 520, 20, 0, 0, 1, 41973, 4542, 120, 20, 0, 0, 1, 776250, 46550, 672, 70, 0, 0, 0, 1, 16231381, 540136, 5516, 112, 70, 0, 0, 0, 1, 380333228, 7045020, 40140, 1848, 252, 0, 0, 0, 0, 1 (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..140, flattened FORMULA E.g.f. of column k=0: 1 +1/(1+LambertW(-x)) -1/(2-exp(x)); e.g.f. of column k>0: 1/(1-Sum_{j>=k} x^j/j!) - 1/(1-Sum_{j>=k+1} x^j/j!). T(n,k) = A245732(n,k) - A245732(n,k+1). EXAMPLE T(2,0) = 1: (2,2). T(2,1) = 2: (1,2), (2,1). T(2,2) = 1: (1,1). T(3,1) = 12: (1,1,2), (1,2,1), (1,2,2), (1,2,3), (1,3,2), (2,1,1), (2,1,2), (2,1,3), (2,2,1), (2,3,1), (3,1,2), (3,2,1). T(3,3) = 1: (1,1,1). Triangle T(n,k) begins: 0 : 1; 1 : 0, 1; 2 : 1, 2, 1; 3 : 14, 12, 0, 1; 4 : 181, 68, 6, 0, 1; 5 : 2584, 520, 20, 0, 0, 1; 6 : 41973, 4542, 120, 20, 0, 0, 1; 7 : 776250, 46550, 672, 70, 0, 0, 0, 1; 8 : 16231381, 540136, 5516, 112, 70, 0, 0, 0, 1; ... MAPLE b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)*binomial(n, j), j=k..n)) end: g:= (n, k)-> `if`(k=0, n^n, `if`(n=0, 0, b(n, k))): T:= (n, k)-> g(n, k) -g(n, k+1): seq(seq(T(n, k), k=0..n), n=0..12); MATHEMATICA b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n-j, k]*Binomial[n, j], {j, k, n}]]; g[n_, k_] := If[k == 0, n^n, If[n == 0, 0, b[n, k]]]; T[n_, k_] := g[n, k] - g[n, k+1]; T[0, 0] = 1; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 27 2015, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A133286 (for n>0), A245854, A245855, A245856, A245857, A245858, A245859, A245860, A245861, A245862, A245863. Row sums give A000312. T(2n,n) gives A000984(n). Cf. A245732. Sequence in context: A307804 A338207 A063613 * A080346 A216445 A124026 Adjacent sequences: A245730 A245731 A245732 * A245734 A245735 A245736 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jul 30 2014 STATUS approved

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Last modified January 29 13:51 EST 2023. Contains 359923 sequences. (Running on oeis4.)