A162299 Faulhaber's triangle: triangle T(k,y) read by rows, giving denominator of the coefficient [m^y] of the polynomial Sum_{x=1..m} x^(k-1).

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

Offset: 0

Author

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

Keywords

• nonn
• tabl
• frac

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

Examples

			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;
  ...
		

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

Crossrefs

Cf. A000367, A162298 (numerators).

See also A220962/A220963.

Cf. A053382, A053383.

Programs

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 *)

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.

Extensions

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