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).
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
Offset: 0
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;
...
Links
- Alois P. Heinz, Rows n = 0..140, flattened
- R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1991 (Seventh printing).Second ed. 1994.
- R. Remmert, Funktionentheorie I, Zweite Auflage, Springer-Verlag, 1989. English version: Classical topics in complex function theory, Springer, 1998.
- Mohammad Torabi-Dashti, Faulhaber's Triangle, College Math. J., Vol. 42, No. 2 (2011), pp. 96-97.
- Mohammad Torabi-Dashti, Faulhaber’s Triangle. [Annotated scanned copy of preprint]
- Eric Weisstein's MathWorld, Power Sum.
Programs
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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
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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] // Numerator, {n, 0, 13}, {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.
T(k,m)= numerator(r(k,m)) with r(k,m)= 1/(k+1) if m=k+1, 1/2 if m=k, and (B(k+1-m)/(k+1-m))*binomial(k,m) if m = 1,...,k-1, with the Bernoulli numbers B(n)=A027641(n)/A027642(n). Alternatively r(k,m) = ((-1)^(k+1-m))*sum(S(k,l)*s(l+1,m)/(l+1),l=(m-1),...,k), k>=0, m=1,...,k+1, with S given in A048993, and s given in A048994. - Wolfdieter Lang, Oct 23 2011
Extensions
Offset set to 0 by Alois P. Heinz, Feb 19 2021
Comments