OFFSET
0,9
COMMENTS
Row n gives the coefficients which express the sums of the n-th powers of the integers as a linear combination of binomial coefficients, thus:
Sum_{k=1..r} k^n = A103438(n+r,r) = Sum_{k=0..n} T(n+1,k) * C(r,k),
where, by convention, C(r,k) = 0 whenever r < k. - Robert B Fowler, Jan 16 2023
LINKS
Peter Luschny, Row 0 to 44, recomputed Jan 19 2019
FORMULA
Row n gives the coefficients of the polynomial defined by p(x, 0) = 1 and for n > 0 p(x, n) = Sum_{k >= 0} k^(n-1)*(x/(1 + x))^k = PolyLog(-n+1, x/(1+x)).
T(n, k) = (k - 1)! * Stirling2(n, k) for k > 0. - Detlef Meya, Jan 06 2024
EXAMPLE
Triangle begins:
1;
0, 1;
0, 1, 1;
0, 1, 3, 2;
0, 1, 7, 12, 6;
0, 1, 15, 50, 60, 24;
0, 1, 31, 180, 390, 360, 120;
0, 1, 63, 602, 2100, 3360, 2520, 720;
0, 1, 127, 1932, 10206, 25200, 31920, 20160, 5040;
...
MAPLE
CL := (f, x) -> PolynomialTools:-CoefficientList(f, x):
A142071row := proc(n) 1 - log(1 - x*(exp(z) - 1)):
series(%, z, 12): CL(n!*coeff(%, z, n), x) end:
for n from 0 by 1 to 7 do A142071row(n) od;
# Alternative:
A142071Row := proc(n) if n=0 then [1] else
CL(convert(series(polylog(-n+1, z/(1+z)), z, n*2), polynom), z) fi end:
seq(A142071Row(n), n=0..6); # Peter Luschny, Sep 06 2018
MATHEMATICA
T[n_, k_] := If[k==0, Floor[1/(n + 1)], (k - 1)!*StirlingS2[n, k]]; Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Detlef Meya, Jan 06 2024 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula and Gary W. Adamson, Sep 15 2008
EXTENSIONS
Edited, T(0,0) = 1 prepended and new name by Peter Luschny, Sep 06 2018
STATUS
approved