A159688 Triangle read by rows, denominators of Jakob Bernoulli's "Sums of Powers" triangle.

1, 2, 2, 3, 2, 6, 4, 2, 4, 5, 2, 3, -30, 6, 2, 12, -12, 7, 2, 2, -6, 42, 8, 2, 12, -24, 12, 9, 2, 3, -15, 9, -30, 10, 2, 4, -10, 2, -20, 11, 2, 6, -1, 1, -2, 66, 12, 2, 12, -8, 6, -8, 12, 13, 2, 1, -6, 7, -10, 3, -2730, 14, 2, 12, -60, 28, -20, 12, -420, 15, 2, 6, -30, 18, -10, 6, -90, 6, 16, 2, 4, -24, 12, -16, 12, -24, 4

Offset: 0

Author

Gary W. Adamson, Apr 19 2009

Keywords

• tabf
• sign

Comments

Let the triangle = T. Row sums = 1. Row sums of n-th binomial transform of T = powers of (n-1). Then multiply the results by the partial sum operator, (1; 1,1; 1,1,1; ...) to obtain Bernoulli's "Sums of Powers".

Inserting zeros to account for (n+1) terms per row, right border = Bernoulli numbers: (A106458): (1, 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ...).

Examples

			Let row 0 = 1; followed by the corrected table, giving denominators:
   1;
   2, 2;
   3, 2,  6;
   4, 2,  4;
   5, 2,  3, -30;
   6, 2, 12, -12;
   7, 2,  2,  -6, 42;
   8, 2, 12, -24, 12;
   9, 2,  3, -15,  9, -30;
  10, 2,  4, -10,  2, -20;
  11, 2,  6,  -1,  1,  -2, 66;
  ...
The complete triangle with row 0 = 1, along with numerators:
  1;
  1/2,  1/2;
  1/3,  1/2, 1/6;
  1/4,  1/2, 1/4;
  1/5,  1/2, 1/3,  -1/30;
  1/6,  1/2, 5/12, -1/12;
  1/7,  1/2, 1/2,  -1/6,  1/42;
  1/8,  1/2, 7/12, -7/14, 1/12;
  1/9,  1/2, 2/3,  -7/15, 1/2, -3/20;
  1/10, 1/2, 3/4,  -7/10, 1/2, -3/20;
  1/11, 1/2, 5/6,  -1/1,  1/1, -1/2,  5/66;
  ...
		

References

• Jakob Bernoulli, "Ars conjectandi", posthumously published in 1713, in which Bernoulli gives the table "Summae Potestatum (Sums of Powers) [cf. Young, p. 86].
• Robert M. Young, "Excursions in Calculus", MAA, 1992.

Links

• Seiichi Manyama, Rows n = 0..200, flattened
• Wikipedia, Faulhaber's formula.

Crossrefs

Cf. A106458.

Programs

Mathematica

f[n_, x_] := f[n, x] = ((x+1)^(n+1) - 1)/(n+1) - Sum[Binomial[n+1, k]*f[k, x], {k, 0, n-1}]/(n+1); f[0, x_] := x; row[n_] := CoefficientList[f[n, x], x] // Reverse // (Sign[#]*Denominator[#])& // DeleteCases[#,0]&; Table[row[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Dec 29 2012 *)

Extensions

Extended to 15 rows by Jean-François Alcover, Dec 29 2012