A162298 - OEIS

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

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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/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):

1/2, 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).