login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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). 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,19

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..105.

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;

  ...

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

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 23 11:24 EDT 2019. Contains 326222 sequences. (Running on oeis4.)