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%I #29 Feb 28 2023 17:02:40
%S 1,1,1,1,2,1,1,5,5,1,1,12,22,12,1,1,27,92,92,27,1,1,58,359,604,359,58,
%T 1,1,121,1311,3607,3607,1311,121,1,1,248,4540,19912,31238,19912,4540,
%U 248,1,1,503,15110,102842,244424,244424,102842,15110,503,1,1,1014
%N Triangle of Eulerian numbers with rows multiplied by 1 + x.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
%H Alois P. Heinz, <a href="/A008518/b008518.txt">Rows n = 0..140, flattened</a>
%H Huyile Liang, Yanni Pei, and Yi Wang, <a href="https://arxiv.org/abs/2302.11856">Analytic combinatorics of coordination numbers of cubic lattices</a>, arXiv:2302.11856 [math.CO], 2023. See p. 22.
%F E.g.f.: (exp(x) - y*exp(y*x))/(exp(y*x) - y*exp(x)). - _Vladeta Jovovic_, Apr 06 2001
%F T(n,k) = A123125(n,k) + A123125(n,k+1), with A123125(n,n+1) = 0. - _Franck Maminirina Ramaharo_, Oct 21 2018
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 5, 5, 1;
%e 1, 12, 22, 12, 1;
%e 1, 27, 92, 92, 27, 1;
%e 1, 58, 359, 604, 359, 58, 1;
%e 1, 121, 1311, 3607, 3607, 1311, 121, 1;
%e ...
%t t[n_ /; n >= 0, 0] = 1; t[n_, k_] /; k<0 || k>n = 0; t[n_, k_] := t[n, k] = (n-k) t[n-1, k-1] + (k+1) t[n-1, k];
%t A[n_, k_] /; k == n+1 = 0; A[n_, k_] := t[n, n-k];
%t T[n_, k_] := A[n, k] + A[n, k+1];
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 26 2019, after _Franck Maminirina Ramaharo_ *)
%Y Cf. A008292.
%Y Columns include A000325. Row sums are in A098558.
%Y T(2n,n) gives A177042.
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_.
%E More terms from _Vladeta Jovovic_, Apr 06 2001
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