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%I #39 Aug 04 2022 10:19:18
%S 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,
%T 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,
%U 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
%N 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).
%C 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
%H Alois P. Heinz, <a href="/A162299/b162299.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., 42:2 (2011), 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 ...
%e The triangle starts in row k=1 with columns 1<=y<=k as
%e 1
%e 2 2
%e 6 2 3
%e 1 4 2 4
%e 30 1 3 2 5
%e 1 12 1 12 2 6
%e 42 1 6 1 2 2 7
%e 1 12 1 24 1 12 2 8
%e 30 1 9 1 15 1 3 2 9
%e 1 20 1 2 1 10 1 4 2 10
%e 66 1 2 1 1 1 1 1 6 2 11
%e 1 12 1 8 1 6 1 8 1 12 2 12
%e 2730 1 3 1 10 1 7 1 6 1 1 2 13
%e 1 420 1 12 1 20 1 28 1 60 1 12 2 14
%e 6 1 90 1 6 1 10 1 18 1 30 1 6 2 15
%e ...
%e Initial rows of triangle of fractions:
%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 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
%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] // Denominator, {n, 0, 14}, {k, 1, n + 1}] // Flatten (* _Jean-François Alcover_, Aug 04 2022 *)
%Y Cf. A000367, A162298 (numerators).
%Y See also A220962/A220963.
%Y Cf. A053382, A053383.
%K nonn,tabl,frac
%O 0,2
%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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