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A354977 Triangle read by rows. T(n, k) = Sum_{j=0..n}((-1)^(n-j)*binomial(n, j)*j^(n+k)) / n!.

%I A354977 #14 Feb 25 2025 10:35:41
%S A354977 1,1,1,1,3,7,1,6,25,90,1,10,65,350,1701,1,15,140,1050,6951,42525,1,21,
%T A354977 266,2646,22827,179487,1323652,1,28,462,5880,63987,627396,5715424,
%U A354977 49329280,1,36,750,11880,159027,1899612,20912320,216627840,2141764053
%N A354977 Triangle read by rows. T(n, k) = Sum_{j=0..n}((-1)^(n-j)*binomial(n, j)*j^(n+k)) / n!.
%H A354977 Andrew Francis and Michael Hendriksen, <a href="https://arxiv.org/abs/2502.14223">Counting spinal phylogenetic networks</a>, arXiv:2502.14223 [q-bio.PE], 2025. See p. 13.
%F A354977 T(n, k) = Stirling2(n + k, n).
%e A354977 Triangle T(n, k) begins:
%e A354977 [0] 1;
%e A354977 [1] 1,  1;
%e A354977 [2] 1,  3,   7;
%e A354977 [3] 1,  6,  25,    90;
%e A354977 [4] 1, 10,  65,   350,   1701;
%e A354977 [5] 1, 15, 140,  1050,   6951,   42525;
%e A354977 [6] 1, 21, 266,  2646,  22827,  179487,  1323652;
%e A354977 [7] 1, 28, 462,  5880,  63987,  627396,  5715424,  49329280;
%e A354977 [8] 1, 36, 750, 11880, 159027, 1899612, 20912320, 216627840, 2141764053;
%p A354977 T := (n, k) -> add((-1)^(n - j)*binomial(n, j)*j^(n + k), j = 0..n) / n!:
%p A354977 seq(seq(T(n, k), k = 0..n), n = 0..8);
%Y A354977 T(n,1) = A000217, T(n,n) = A007820, A354978 (row sums), A048993.
%K A354977 nonn,tabl
%O A354977 0,5
%A A354977 _Peter Luschny_, Jun 15 2022