A220962 Faulhaber’s triangle: triangle of numerators of coefficients of power-sum polynomials.

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, -1, 0, 1, 1, 1, 0, 0, -1, 0, 5, 1, 1, 0, 1, 0, -1, 0, 1, 1, 1, 0, 0, 1, 0, -7, 0, 7, 1, 1, 0, -1, 0, 2, 0, -7, 0, 2, 1, 1, 0, 0, -3, 0, 1, 0, -7, 0, 3, 1, 1, 0, 5, 0, -1, 0, 1, 0, -1, 0, 5, 1, 1

Offset

0,25

Comments

This version of Faulhaber's triangle, A220962/A220963, is essentially the same as A162298/A162299 except for having an extra column of 0's. See A162298/A162299 for further information. - N. J. A. Sloane, Jan 28 2017

Links

Table of n, a(n) for n=0..76.

Mohammad Torabi-Dashti, Faulhaber’s Triangle [Annotated scanned copy of preprint]

Mohammad Torabi-Dashti, Faulhaber's Triangle, College Math. J., 42:2 (2011), 96-97.

Eric Weisstein's MathWorld, Power Sum

Example

Rows start:
0,1;
0,1,1;
0,1,1,1;
0,0,1,1,1;
0,-1,0,1,1,1;
0,0,-1,0,5,1,1;
0,1,0,-1,0,1,1,1;
0,0,1,0,-7,0,7,1,1;
0,-1,0,2,0,-7,0,2,1,1;
...

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] // Numerator; Table[row[n], {n, 0, 10}] // Flatten

Crossrefs

Cf. A220963 (denominators).

See also A162298/A162299.

Sequence in context: A112991 A346081 A137373 * A201292 A348975 A271343

Adjacent sequences: A220959 A220960 A220961 * A220963 A220964 A220965

Keyword

sign,tabf,frac

Author

Jean-François Alcover, Dec 27 2012

Status

approved