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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). 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

Table of n, a(n) for n=1..120.

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

CROSSREFS

Cf. A000367, A162299 (denominators).

See also A220962/A220963.

Cf. A000367, A162298 (numerators), A053382, A053383.

Sequence in context: A205030 A278250 A134339 * A281552 A205506 A110141

Adjacent sequences:  A162296 A162297 A162298 * A162300 A162301 A162302

KEYWORD

nonn,tabl,frac

AUTHOR

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

STATUS

approved

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Last modified June 6 17:28 EDT 2020. Contains 334830 sequences. (Running on oeis4.)