OFFSET
0,2
COMMENTS
There are many versions of Faulhaber's triangle: search the OEIS for his name. For example, A220862/A220963 is essentially the same as this triangle, except for an initial column of 0's. - N. J. A. Sloane, Jan 28 2017
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
Mohammad Torabi-Dashti, Faulhaber’s Triangle, College Math. J., 42:2 (2011), 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;
...
The triangle starts in row k=1 with columns 1<=y<=k as
1
2 2
6 2 3
1 4 2 4
30 1 3 2 5
1 12 1 12 2 6
42 1 6 1 2 2 7
1 12 1 24 1 12 2 8
30 1 9 1 15 1 3 2 9
1 20 1 2 1 10 1 4 2 10
66 1 2 1 1 1 1 1 6 2 11
1 12 1 8 1 6 1 8 1 12 2 12
2730 1 3 1 10 1 7 1 6 1 1 2 13
1 420 1 12 1 20 1 28 1 60 1 12 2 14
6 1 90 1 6 1 10 1 18 1 30 1 6 2 15
...
Initial rows of triangle of fractions:
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
A162299 := proc(k, y) local gf, x; gf := sum(x^(k-1), x=1..m) ; coeftayl(gf, m=0, y) ; denom(%) ; end proc: # R. J. Mathar, Jan 24 2011
# 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] // Denominator, {n, 0, 14}, {k, 1, n + 1}] // Flatten (* Jean-François Alcover, Aug 04 2022 *)
CROSSREFS
KEYWORD
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