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A159688 Triangle read by rows, denominators of Jakob Bernoulli's "Sums of Powers" triangle.

%I A159688 #22 Feb 04 2023 14:15:23
%S A159688 1,2,2,3,2,6,4,2,4,5,2,3,-30,6,2,12,-12,7,2,2,-6,42,8,2,12,-24,12,9,2,
%T A159688 3,-15,9,-30,10,2,4,-10,2,-20,11,2,6,-1,1,-2,66,12,2,12,-8,6,-8,12,13,
%U A159688 2,1,-6,7,-10,3,-2730,14,2,12,-60,28,-20,12,-420,15,2,6,-30,18,-10,6,-90,6,16,2,4,-24,12,-16,12,-24,4
%N A159688 Triangle read by rows, denominators of Jakob Bernoulli's "Sums of Powers" triangle.
%C A159688 Let the triangle = T. Row sums = 1. Row sums of n-th binomial transform of T = powers of (n-1). Then multiply the results by the partial sum operator, (1; 1,1; 1,1,1; ...) to obtain Bernoulli's "Sums of Powers".
%C A159688 Inserting zeros to account for (n+1) terms per row, right border = Bernoulli numbers: (A106458): (1, 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ...).
%D A159688 Jakob Bernoulli, "Ars conjectandi", posthumously published in 1713, in which Bernoulli gives the table "Summae Potestatum (Sums of Powers) [cf. Young, p. 86].
%D A159688 Robert M. Young, "Excursions in Calculus", MAA, 1992.
%H A159688 Seiichi Manyama, <a href="/A159688/b159688.txt">Rows n = 0..200, flattened</a>
%H A159688 Wikipedia, <a href="https://en.wikipedia.org/wiki/Faulhaber%27s_formula">Faulhaber's formula</a>.
%e A159688 Let row 0 = 1; followed by the corrected table, giving denominators:
%e A159688    1;
%e A159688    2, 2;
%e A159688    3, 2,  6;
%e A159688    4, 2,  4;
%e A159688    5, 2,  3, -30;
%e A159688    6, 2, 12, -12;
%e A159688    7, 2,  2,  -6, 42;
%e A159688    8, 2, 12, -24, 12;
%e A159688    9, 2,  3, -15,  9, -30;
%e A159688   10, 2,  4, -10,  2, -20;
%e A159688   11, 2,  6,  -1,  1,  -2, 66;
%e A159688   ...
%e A159688 The complete triangle with row 0 = 1, along with numerators:
%e A159688   1;
%e A159688   1/2,  1/2;
%e A159688   1/3,  1/2, 1/6;
%e A159688   1/4,  1/2, 1/4;
%e A159688   1/5,  1/2, 1/3,  -1/30;
%e A159688   1/6,  1/2, 5/12, -1/12;
%e A159688   1/7,  1/2, 1/2,  -1/6,  1/42;
%e A159688   1/8,  1/2, 7/12, -7/14, 1/12;
%e A159688   1/9,  1/2, 2/3,  -7/15, 1/2, -3/20;
%e A159688   1/10, 1/2, 3/4,  -7/10, 1/2, -3/20;
%e A159688   1/11, 1/2, 5/6,  -1/1,  1/1, -1/2,  5/66;
%e A159688   ...
%t A159688 f[n_, x_] := f[n, x] = ((x+1)^(n+1) - 1)/(n+1) - Sum[Binomial[n+1, k]*f[k, x], {k, 0, n-1}]/(n+1); f[0, x_] := x; row[n_] := CoefficientList[f[n, x], x] // Reverse // (Sign[#]*Denominator[#])& // DeleteCases[#,0]&; Table[row[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Dec 29 2012 *)
%Y A159688 Cf. A106458.
%K A159688 tabf,sign
%O A159688 0,2
%A A159688 _Gary W. Adamson_, Apr 19 2009
%E A159688 Extended to 15 rows by _Jean-François Alcover_, Dec 29 2012