A008518 - OEIS

%I #33 Jun 19 2024 02:03:27

%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

%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

%F From _G. C. Greubel_, Jun 18 2024: (Start)

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

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

%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_ *)

%o (Magma)

%o Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >;

%o [Eulerian(n, k-1) + Eulerian(n, k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Jun 18 2024

%o (SageMath)

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

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

%Y Columns include A000325 (k=1).

%K nonn,tabl

%O 0,5

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Apr 06 2001