A142071 Expansion of the exponential generating function 1 - log(1 - x*(exp(z) - 1)), triangle read by rows, T(n,k) for n >= 0 and 0 <= k <= n.

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, 0, 1, 255, 6050, 46620, 166824, 317520, 332640, 181440, 40320, 0, 1, 511, 18660

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

Column k = 0 is A000007.

Cf. A028246, A163626, A000629 (row sums).

Cf. A103438, A007318 (binomial coefficients).

Sequence in context: A154477 A363674 A322324 * A350448 A291680 A193283

Adjacent sequences: A142068 A142069 A142070 * A142072 A142073 A142074

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