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A008518 Triangle of Eulerian numbers with rows multiplied by 1 + x.

%I A008518 #33 Jun 19 2024 02:03:27
%S A008518 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 A008518 1,1,121,1311,3607,3607,1311,121,1,1,248,4540,19912,31238,19912,4540,
%U A008518 248,1,1,503,15110,102842,244424,244424,102842,15110,503,1
%N A008518 Triangle of Eulerian numbers with rows multiplied by 1 + x.
%D A008518 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
%D A008518 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.
%D A008518 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
%H A008518 Alois P. Heinz, <a href="/A008518/b008518.txt">Rows n = 0..140, flattened</a>
%H A008518 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 A008518 E.g.f.: (exp(x) - y*exp(y*x))/(exp(y*x) - y*exp(x)). - _Vladeta Jovovic_, Apr 06 2001
%F A008518 T(n,k) = A123125(n,k) + A123125(n,k+1), with A123125(n,n+1) = 0. - _Franck Maminirina Ramaharo_, Oct 21 2018
%F A008518 From _G. C. Greubel_, Jun 18 2024: (Start)
%F A008518 T(n, n-k) = T(n, k).
%F A008518 Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)
%e A008518 Triangle begins:
%e A008518    1;
%e A008518    1,   1;
%e A008518    1,   2,    1;
%e A008518    1,   5,    5,    1;
%e A008518    1,  12,   22,   12,    1;
%e A008518    1,  27,   92,   92,   27,    1;
%e A008518    1,  58,  359,  604,  359,   58,   1;
%e A008518    1, 121, 1311, 3607, 3607, 1311, 121, 1;
%e A008518    ...
%t A008518 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 A008518 A[n_, k_] /; k == n+1 = 0; A[n_, k_] := t[n, n-k];
%t A008518 T[n_, k_] := A[n, k] + A[n, k+1];
%t A008518 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 26 2019, after _Franck Maminirina Ramaharo_ *)
%o A008518 (Magma)
%o A008518 Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >;
%o A008518 [Eulerian(n, k-1) + Eulerian(n, k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Jun 18 2024
%o A008518 (SageMath)
%o A008518 def Eulerian(n,k): return sum((-1)^j*binomial(n+1, j)*(k-j+1)^n for j in range(k+2))
%o A008518 flatten([[Eulerian(n,k-1) + Eulerian(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jun 18 2024
%Y A008518 Cf. A000007, A008292, A098558 (row sums), A177042 (T(2*n, n)).
%Y A008518 Columns include A000325 (k=1).
%K A008518 nonn,tabl
%O A008518 0,5
%A A008518 _N. J. A. Sloane_
%E A008518 More terms from _Vladeta Jovovic_, Apr 06 2001