A162298 Faulhaber's triangle: triangle T(k,y) read by rows, giving numerator of the coefficient [m^y] of the polynomial Sum_{x=1..m} x^(k-1).

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, 0, 1, 1, 1, 0, -1, 0, 5, 1, 1, 1, 0, -1, 0, 1, 1, 1, 0, 1, 0, -7, 0, 7, 1, 1, -1, 0, 2, 0, -7, 0, 2, 1, 1, 0, -3, 0, 1, 0, -7, 0, 3, 1, 1, 5, 0, -1, 0, 1, 0, -1, 0, 5, 1, 1, 0, 5, 0, -11, 0, 11, 0, -11, 0, 11, 1, 1, -691, 0, 5, 0, -33, 0, 22, 0, -11, 0, 1, 1, 1, 0, -691, 0, 65, 0, -143, 0, 143, 0, -143, 0, 13, 1, 1

Offset

0,19

Comments

There are many versions of Faulhaber's triangle: search the OEIS for his name. For example, A220962/A220963 is essentially the same as this triangle, except for an initial column of 0's. - N. J. A. Sloane, Jan 28 2017

Named after the German mathematician Johann Faulhaber (1580-1653). - Amiram Eldar, Jun 13 2021

Links

Alois P. Heinz, Rows n = 0..140, flattened

Mohammad Torabi-Dashti, Faulhaber's Triangle, College Math. J., Vol. 42, No. 2 (2011), pp. 96-97.

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

Eric Weisstein's MathWorld, Power Sum.

Formula

Faulhaber's triangle of fractions H(n,k) (n >= 0, 1 <= k <= n+1) is defined by: H(0,1)=1; for 2<=k<=n+1, H(n,k) = (n/k)*H(n-1,k-1) with H(n,1) = 1 - Sum_{i=2..n+1}H(n,i). - N. J. A. Sloane, Jan 28 2017

Sum_{x=1..m} x^(k-1) = (Bernoulli(k,m+1)-Bernoulli(k))/k.

Example

The first few polynomials:
    m;
   m/2  + m^2/2;
   m/6  + m^2/2 + m^3/3;
    0   + m^2/4 + m^3/2 + m^4/4;
  -m/30 +   0   + m^3/3 + m^4/2 + m^5/5;
  ...
Initial rows of Faulhaber's triangle of fractions H(n, k) (n >= 0, 1 <= k <= n+1):
    1;
   1/2,  1/2;
   1/6,  1/2,  1/3;
    0,   1/4,  1/2,  1/4;
  -1/30,  0,   1/3,  1/2,  1/5;
    0,  -1/12,  0,   5/12, 1/2,  1/6;
   1/42,  0,  -1/6,   0,   1/2,  1/2,  1/7;
    0,   1/12,  0,  -7/24,  0,   7/12, 1/2,  1/8;
  -1/30,  0,   2/9,   0,  -7/15,  0,   2/3,  1/2,  1/9;
  ...

Maple

A162298 := proc(k, y) local gf, x; gf := sum(x^(k-1), x=1..m) ; coeftayl(gf, m=0, y) ; numer(%) ; end proc: # R. J. Mathar, Mar 26 2013
# To produce Faulhaber's triangle of fractions H(n, k) (n >= 0, 1 <= k <= n+1):
H:=proc(n, k) option remember; local i;
if n<0 or k>n+1 then 0;
elif n=0 then 1;
elif k>1 then (n/k)*H(n-1, k-1);
else 1 - add(H(n, i), i=2..n+1); fi; end;
for n from 0 to 10 do lprint([seq(H(n, k), k=1..n+1)]); od:
for n from 0 to 12 do lprint([seq(numer(H(n, k)), k=1..n+1)]); od: # A162298
for n from 0 to 12 do lprint([seq(denom(H(n, k)), k=1..n+1)]); od: # A162299 # N. J. A. Sloane, Jan 28 2017

Mathematica

H[n_, k_] := H[n, k] = Which[n < 0 || k > n+1, 0, n == 0, 1, k > 1, (n/k)* H[n-1, k-1], True, 1 - Sum[H[n, i], {i, 2, n+1}]];
Table[H[n, k] // Numerator, {n, 0, 13}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 04 2022 *)

Crossrefs

Cf. A000367, A162299 (denominators).

See also A220962/A220963.

Sequence in context: A227577 A281446 A196840 * A196755 A199510 A146306

Adjacent sequences: A162295 A162296 A162297 * A162299 A162300 A162301

Keyword

tabl,frac,sign

Author

Juri-Stepan Gerasimov, Jun 30 2009 and Jul 02 2009

Extensions

Offset set to 0 by Alois P. Heinz, Feb 19 2021

Status

approved