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A220962 Faulhaber’s triangle: triangle of numerators of coefficients of power-sum polynomials.

%I A220962 #28 Feb 16 2025 08:33:18
%S A220962 0,1,0,1,1,0,1,1,1,0,0,1,1,1,0,-1,0,1,1,1,0,0,-1,0,5,1,1,0,1,0,-1,0,1,
%T A220962 1,1,0,0,1,0,-7,0,7,1,1,0,-1,0,2,0,-7,0,2,1,1,0,0,-3,0,1,0,-7,0,3,1,1,
%U A220962 0,5,0,-1,0,1,0,-1,0,5,1,1
%N A220962 Faulhaber’s triangle: triangle of numerators of coefficients of power-sum polynomials.
%C A220962 This version of Faulhaber's triangle, A220962/A220963, is essentially the same as A162298/A162299 except for having an extra column of 0's. See A162298/A162299 for further information. - _N. J. A. Sloane_, Jan 28 2017
%H A220962 Mohammad Torabi-Dashti, <a href="/A162298/a162298.pdf">Faulhaber’s Triangle</a> [Annotated scanned copy of preprint]
%H A220962 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 A220962 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/PowerSum.html">Power Sum</a>
%e A220962 Rows start:
%e A220962 0,1;
%e A220962 0,1,1;
%e A220962 0,1,1,1;
%e A220962 0,0,1,1,1;
%e A220962 0,-1,0,1,1,1;
%e A220962 0,0,-1,0,5,1,1;
%e A220962 0,1,0,-1,0,1,1,1;
%e A220962 0,0,1,0,-7,0,7,1,1;
%e A220962 0,-1,0,2,0,-7,0,2,1,1;
%e A220962 ...
%t A220962 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] // Numerator; Table[row[n], {n, 0, 10}] // Flatten
%Y A220962 Cf. A220963 (denominators).
%Y A220962 See also A162298/A162299.
%K A220962 sign,tabf,frac
%O A220962 0,25
%A A220962 _Jean-François Alcover_, Dec 27 2012