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%I #20 Oct 17 2022 07:07:12
%S 0,0,2,0,2,21,0,2,45,232,0,2,93,784,3005,0,2,189,2536,13825,45936,0,2,
%T 381,7984,61325,264816,818503,0,2,765,24712,264625,1488096,5623681,
%U 16736896,0,2,1533,75664,1119005,8172576,38025127,132766208,387057609,0
%N Triangle read by rows: T(n,k) = number of functions from an n-element set into but not onto a k-element set.
%H Mohammad K. Azarian, <a href="https://doi.org/10.12988/imf.2022.912321">Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions</a>, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Theorem 2.2(v).
%H D. P. Walsh, <a href="http://capone.mtsu.edu/dwalsh/NONSURJ.pdf">A note on non-surjective functions from [n] to [k]</a>.
%F T(n,k) = A089072(n,k) - A019538(n,k).
%F T(n,k) = Sum_{j=1..k} (-1)^(j-1)*C(k,j)*(k-j)^n. - _Dennis P. Walsh_, Apr 13 2016
%F T(n,k) = k^n - k!*Stirling2(n,k). - _Dennis P. Walsh_, Apr 13 2016
%e T(3,3) = #(functions into) - #(functions onto) = 3^3 - 6 = 21
%e Triangle T(n,k) begins:
%e 0,
%e 0, 2;
%e 0, 2, 21;
%e 0, 2, 45, 232;
%e 0, 2, 93, 784, 3005;
%e 0, 2, 189, 2536, 13825, 45936;
%e 0, 2, 381, 7984, 61325, 264816, 818503;
%e 0, 2, 765, 24712, 264625, 1488096, 5623681, 16736896;
%e 0, 2, 1533, 75664, 1119005, 8172576, 38025127, 132766208, 387057609;
%p T:=(n, k)->sum((-1)^(j-1)*binomial(k, j)*(k-j)^n, j=1..k);
%p seq(seq(T(n, k), k=1..n), n=1..15); # _Dennis P. Walsh_, Apr 13 2016
%Y Cf. A199656, A036679 (diagonal).
%K nonn,tabl,easy
%O 1,3
%A _Clark Kimberling_, Nov 26 2004
%E Offset corrected from 0 to 1 by _Dennis P. Walsh_, Apr 13 2016
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