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A341101
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.
1
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, 240, 64, 0, 720, 2604, 4046, 3570, 1960, 672, 128, 0, 5040, 18828, 30604, 28777, 17360, 6944, 1792, 256, 0, 40320, 154944, 261656, 259056, 167874, 74592, 22848, 4608, 512
OFFSET
0,3
LINKS
Özmen, N., Erkuş-Duman, E. (2019). On the Generalized Sylvester Polynomials. 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.
FORMULA
Sum_{k=0..n-1} T(n, k) = Sum_{k=0..n} binomial(n, k)*(k! - 1) = A097204(n).
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
From Peter Luschny, Nov 24 2022: (Start)
T(n, k) = [x^k] (x^n)*hypergeom([-n, x], [], -1/x).
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)
EXAMPLE
Triangle starts:
n\k 0 1 2 3 4 5 6 7 8 ...
0: 1
1: 0 2
2: 0 1 4
3: 0 2 6 8
4: 0 6 19 24 16
5: 0 24 80 110 80 32
6: 0 120 418 615 500 240 64
7: 0 720 2604 4046 3570 1960 672 128
8: 0 5040 18828 30604 28777 17360 6944 1792 256
MAPLE
T := (n, k) -> add(binomial(n, k - j)*Stirling1(n - k + j, j)*(-1)^(n-k), j=0..k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..9);
# Alternative:
SP := (n, x) -> (x^n)*hypergeom([-n, x], [], -1/x):
row := n -> seq(coeff(simplify(SP(n, x)), x, k), k = 0..n):
for n from 0 to 8 do row(n) od; # Peter Luschny, Nov 23 2022
MATHEMATICA
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 *)
PROG
(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
(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 */
(Python)
from math import factorial
from sympy import Symbol, Poly
x = Symbol("x")
def Coeffs(p) -> list[int]:
return list(reversed(Poly(p, x).all_coeffs()))
def L(n, m, x):
if n == 0:
return 1
if n == 1:
return 1 - m - 2*x
return ((2 * (n - x) - m - 1) * L(n - 1, m, x) / n
- (n - x - m - 1) * L(n - 2, m, x) / n)
def Sylvester(n):
return (-1)**n * factorial(n) * L(n, n, x)
for n in range(7):
print(Coeffs(Sylvester(n))) # Peter Luschny, Dec 13 2022
CROSSREFS
Alternating row sums: (-1)^n*(n+1) = A181983(n+1).
Cf. A000522 (row sums), A097204 (row sums - 2^n), A002627 (row sums - n!).
Sequence in context: A308628 A181670 A261251 * A342909 A081265 A039991
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Feb 09 2021
STATUS
approved