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

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, 1, 1, 121, 1311, 3607, 3607, 1311, 121, 1, 1, 248, 4540, 19912, 31238, 19912, 4540, 248, 1, 1, 503, 15110, 102842, 244424, 244424, 102842, 15110, 503, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl

Examples

			Triangle begins:
   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,   1;
   1, 121, 1311, 3607, 3607, 1311, 121, 1;
   ...
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.
• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 22.

Crossrefs

Cf. A000007, A008292, A098558 (row sums), A177042 (T(2*n, n)).

Columns include A000325 (k=1).

Programs

Magma

Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >;
[Eulerian(n, k-1) + Eulerian(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jun 18 2024

Mathematica

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];
A[n_, k_] /; k == n+1 = 0; A[n_, k_] := t[n, n-k];
T[n_, k_] := A[n, k] + A[n, k+1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2019, after Franck Maminirina Ramaharo *)

SageMath

def Eulerian(n,k): return sum((-1)^j*binomial(n+1, j)*(k-j+1)^n for j in range(k+2))
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

Formula

E.g.f.: (exp(x) - y*exp(y*x))/(exp(y*x) - y*exp(x)). - Vladeta Jovovic, Apr 06 2001

T(n,k) = A123125(n,k) + A123125(n,k+1), with A123125(n,n+1) = 0. - Franck Maminirina Ramaharo, Oct 21 2018

From G. C. Greubel, Jun 18 2024: (Start)

T(n, n-k) = T(n, k).

Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)

Extensions

More terms from Vladeta Jovovic, Apr 06 2001