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A162299 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).

%I A162299 #40 Feb 16 2025 08:33:10
%S A162299 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 A162299 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 A162299 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 A162299 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 A162299 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 A162299 Alois P. Heinz, <a href="/A162299/b162299.txt">Rows n = 0..140, flattened</a>
%H A162299 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 A162299 Mohammad Torabi-Dashti, <a href="/A162298/a162298.pdf">Faulhaber’s Triangle</a> [Annotated scanned copy of preprint]
%H A162299 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/PowerSum.html">Power Sum</a>
%F A162299 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 A162299 Sum_{x=1..m} x^(k-1) = (Bernoulli(k,m+1)-Bernoulli(k))/k.
%e A162299 The first few polynomials:
%e A162299     m;
%e A162299    m/2  + m^2/2;
%e A162299    m/6  + m^2/2 + m^3/3;
%e A162299     0   + m^2/4 + m^3/2 + m^4/4;
%e A162299   -m/30 +   0   + m^3/3 + m^4/2 + m^5/5;
%e A162299   ...
%e A162299 Initial rows of Faulhaber's triangle of fractions H(n, k) (n >= 0, 1 <= k <= n+1):
%e A162299     1;
%e A162299    1/2,  1/2;
%e A162299    1/6,  1/2,  1/3;
%e A162299     0,   1/4,  1/2,  1/4;
%e A162299   -1/30,  0,   1/3,  1/2,  1/5;
%e A162299     0,  -1/12,  0,   5/12, 1/2,  1/6;
%e A162299    1/42,  0,  -1/6,   0,   1/2,  1/2,  1/7;
%e A162299     0,   1/12,  0,  -7/24,  0,   7/12, 1/2,  1/8;
%e A162299   -1/30,  0,   2/9,   0,  -7/15,  0,   2/3,  1/2,  1/9;
%e A162299   ...
%e A162299 The triangle starts in row k=1 with columns 1<=y<=k as
%e A162299      1
%e A162299      2   2
%e A162299      6   2  3
%e A162299      1   4  2  4
%e A162299     30   1  3  2  5
%e A162299      1  12  1 12  2  6
%e A162299     42   1  6  1  2  2  7
%e A162299      1  12  1 24  1 12  2  8
%e A162299     30   1  9  1 15  1  3  2  9
%e A162299      1  20  1  2  1 10  1  4  2 10
%e A162299     66   1  2  1  1  1  1  1  6  2 11
%e A162299      1  12  1  8  1  6  1  8  1 12  2 12
%e A162299   2730   1  3  1 10  1  7  1  6  1  1  2 13
%e A162299      1 420  1 12  1 20  1 28  1 60  1 12  2 14
%e A162299      6   1 90  1  6  1 10  1 18  1 30  1  6  2 15
%e A162299   ...
%e A162299 Initial rows of triangle of fractions:
%e A162299     1;
%e A162299    1/2, 1/2;
%e A162299    1/6, 1/2,  1/3;
%e A162299     0,  1/4,  1/2,  1/4;
%e A162299   -1/30, 0,   1/3,  1/2,  1/5;
%e A162299     0, -1/12,  0,   5/12, 1/2,  1/6;
%e A162299    1/42, 0,  -1/6,   0,   1/2,  1/2,  1/7;
%e A162299     0,  1/12,  0,  -7/24,  0,   7/12, 1/2,  1/8;
%e A162299   -1/30, 0,   2/9,   0,  -7/15,  0,   2/3,  1/2,  1/9;
%e A162299   ...
%p A162299 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 A162299 # To produce Faulhaber's triangle of fractions H(n,k) (n >= 0, 1 <= k <= n+1):
%p A162299 H:=proc(n,k) option remember; local i;
%p A162299 if n<0 or k>n+1 then 0;
%p A162299 elif n=0 then 1;
%p A162299 elif k>1 then (n/k)*H(n-1,k-1);
%p A162299 else 1 - add(H(n,i),i=2..n+1); fi; end;
%p A162299 for n from 0 to 10 do lprint([seq(H(n,k),k=1..n+1)]); od:
%p A162299 for n from 0 to 12 do lprint([seq(numer(H(n,k)),k=1..n+1)]); od: # A162298
%p A162299 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 A162299 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 A162299 Table[H[n, k] // Denominator, {n, 0, 14}, {k, 1, n + 1}] // Flatten (* _Jean-François Alcover_, Aug 04 2022 *)
%Y A162299 Cf. A000367, A162298 (numerators).
%Y A162299 See also A220962/A220963.
%Y A162299 Cf. A053382, A053383.
%K A162299 nonn,tabl,frac
%O A162299 0,2
%A A162299 _Juri-Stepan Gerasimov_, Jun 30 2009 and Jul 02 2009
%E A162299 Offset set to 0 by _Alois P. Heinz_, Feb 19 2021