%I #28 Dec 13 2022 08:12:58

%S 1,0,2,0,1,4,0,2,6,8,0,6,19,24,16,0,24,80,110,80,32,0,120,418,615,500,

%T 240,64,0,720,2604,4046,3570,1960,672,128,0,5040,18828,30604,28777,

%U 17360,6944,1792,256,0,40320,154944,261656,259056,167874,74592,22848,4608,512

%N T(n, k) = Sum_{j=0..k} binomial(n, k - j)*Stirling1(n - k + j, j)*(-1)^(n-k). Triangle read by rows, T(n, k) for 0 <= k <= n.

%H Özmen, N., Erkuş-Duman, E. (2019). <a href="https://doi.org/10.1007/978-3-030-04459-6_5">On the Generalized Sylvester Polynomials</a>. In: Lindahl, K., Lindström, T., Rodino, L., Toft, J., Wahlberg, P. (eds) Analysis, Probability, Applications, and Computation. Trends in Mathematics. Birkhäuser, Cham. See page 48.

%F Sum_{k=0..n-1} T(n, k) = Sum_{k=0..n} binomial(n, k)*(k! - 1) = A097204(n).

%F E.g.f. for row polynomials: P(x, z) := Sum_{k>=0} T(n, k) * x^n * z^k/k! = e^(x*z) / (1 - z)^x = 1 + (2*x) * z + (x + 4*x^2) * z^2/2! + ... - _Michael Somos_, Nov 23 2022

%F From _Peter Luschny_, Nov 24 2022: (Start)

%F T(n, k) = [x^k] (x^n)*hypergeom([-n, x], [], -1/x).

%F T(n, k) = [x^k] (-1)^n * n! * L(n, -x - n, x), where L(n, a, x) is the n-th generalized Laguerre polynomial. (End)

%e Triangle starts:

%e n\k 0 1 2 3 4 5 6 7 8 ...

%e 0: 1

%e 1: 0 2

%e 2: 0 1 4

%e 3: 0 2 6 8

%e 4: 0 6 19 24 16

%e 5: 0 24 80 110 80 32

%e 6: 0 120 418 615 500 240 64

%e 7: 0 720 2604 4046 3570 1960 672 128

%e 8: 0 5040 18828 30604 28777 17360 6944 1792 256

%p T := (n, k) -> add(binomial(n, k - j)*Stirling1(n - k + j, j)*(-1)^(n-k), j=0..k):

%p seq(print(seq(T(n,k), k = 0..n)), n = 0..9);

%p # Alternative:

%p SP := (n, x) -> (x^n)*hypergeom([-n, x], [], -1/x):

%p row := n -> seq(coeff(simplify(SP(n, x)), x, k), k = 0..n):

%p for n from 0 to 8 do row(n) od; # _Peter Luschny_, Nov 23 2022

%t T[ n_, k_] := If[ n<0, 0, n! * Coefficient[ SeriesCoefficient[ E^(x * z) / (1 - z)^x, {z, 0, n}], x, k]]; (* _Michael Somos_, Nov 23 2022 *)

%o (PARI) T(n, k) = sum(j=0, k, binomial(n, k-j)*stirling(n-k+j, j, 1)*(-1)^(n-k)); \\ _Michel Marcus_, Feb 11 2021

%o (PARI) {T(n, k) = if( n<0, 0, n! * polcoeff( polcoeff( exp(x*y) / (1 - x + x * O(x^n))^y, n), k))}; /* _Michael Somos_, Nov 23 2022 */

%o (Python)

%o from math import factorial

%o from sympy import Symbol, Poly

%o x = Symbol("x")

%o def Coeffs(p) -> list[int]:

%o return list(reversed(Poly(p, x).all_coeffs()))

%o def L(n, m, x):

%o if n == 0:

%o return 1

%o if n == 1:

%o return 1 - m - 2*x

%o return ((2 * (n - x) - m - 1) * L(n - 1, m, x) / n

%o - (n - x - m - 1) * L(n - 2, m, x) / n)

%o def Sylvester(n):

%o return (-1)**n * factorial(n) * L(n, n, x)

%o for n in range(7):

%o print(Coeffs(Sylvester(n))) # _Peter Luschny_, Dec 13 2022

%Y Alternating row sums: (-1)^n*(n+1) = A181983(n+1).

%Y Cf. A000522 (row sums), A097204 (row sums - 2^n), A002627 (row sums - n!).

%Y Cf. A000166, A340264.

%K nonn,tabl

%O 0,3

%A _Peter Luschny_, Feb 09 2021