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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). 7

%I

%S 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,

%T 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,

%U 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

%N 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).

%C 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

%C Named after the German mathematician Johann Faulhaber (1580-1653). - _Amiram Eldar_, Jun 13 2021

%H Alois P. Heinz, <a href="/A162298/b162298.txt">Rows n = 0..140, flattened</a>

%H Mohammad Torabi-Dashti, <a href="http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/faulhaber-s-triangle">Faulhaber's Triangle</a>, College Math. J., Vol. 42, No. 2 (2011), pp. 96-97.

%H Mohammad Torabi-Dashti, <a href="/A162298/a162298.pdf">Faulhaber’s Triangle</a>. [Annotated scanned copy of preprint]

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/PowerSum.html">Power Sum</a>.

%F 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

%F Sum_{x=1..m} x^(k-1) = (Bernoulli(k,m+1)-Bernoulli(k))/k.

%e The first few polynomials:

%e m;

%e m/2 + m^2/2;

%e m/6 + m^2/2 + m^3/3;

%e 0 + m^2/4 + m^3/2 + m^4/4;

%e -m/30 + 0 + m^3/3 + m^4/2 + m^5/5;

%e ...

%e Initial rows of Faulhaber's triangle of fractions H(n, k) (n >= 0, 1 <= k <= n+1):

%e 1;

%e 1/2, 1/2;

%e 1/6, 1/2, 1/3;

%e 0, 1/4, 1/2, 1/4;

%e -1/30, 0, 1/3, 1/2, 1/5;

%e 0, -1/12, 0, 5/12, 1/2, 1/6;

%e 1/42, 0, -1/6, 0, 1/2, 1/2, 1/7;

%e 0, 1/12, 0, -7/24, 0, 7/12, 1/2, 1/8;

%e -1/30, 0, 2/9, 0, -7/15, 0, 2/3, 1/2, 1/9;

%e ...

%p 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

%p # To produce Faulhaber's triangle of fractions H(n,k) (n >= 0, 1 <= k <= n+1):

%p H:=proc(n,k) option remember; local i;

%p if n<0 or k>n+1 then 0;

%p elif n=0 then 1;

%p elif k>1 then (n/k)*H(n-1,k-1);

%p else 1 - add(H(n,i),i=2..n+1); fi; end;

%p for n from 0 to 10 do lprint([seq(H(n,k),k=1..n+1)]); od:

%p for n from 0 to 12 do lprint([seq(numer(H(n,k)),k=1..n+1)]); od: # A162298

%p 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

%Y Cf. A000367, A162299 (denominators).

%Y See also A220962/A220963.

%K tabl,frac,sign

%O 0,19

%A _Juri-Stepan Gerasimov_, Jun 30 2009 and Jul 02 2009

%E Offset set to 0 by _Alois P. Heinz_, Feb 19 2021

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Last modified August 5 12:52 EDT 2021. Contains 346469 sequences. (Running on oeis4.)