A162298 - OEIS

%I #57 Aug 04 2022 10:19:23

%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

%t 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}]];

%t Table[H[n, k] // Numerator, {n, 0, 13}, {k, 1, n+1}] // Flatten (* _Jean-François Alcover_, Aug 04 2022 *)

%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