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Faulhaber's Formula


In a rare 1631 work entitled Academiae Algebrae, J. Faulhaber published a number of formulae for power sums of the first n positive integers. A detailed analysis of Faulhaber's work may be found in Knuth (1993) and, with a few amendments, in Knuth (2001).

Among the results presented by Faulhaber (without any indication of how they were derived) were the sums of odd powers

sum_(k=1)^(n)k=N
(1)
sum_(k=1)^(n)k^3=N^2
(2)
sum_(k=1)^(n)k^5=1/3(4N^3-N^2)
(3)
sum_(k=1)^(n)k^7=1/3(6N^4-4N^3+N^2)
(4)
sum_(k=1)^(n)k^9=1/5(16N^5-20N^4+12N^3-3N^2)
(5)
sum_(k=1)^(n)k^(11)=1/3(16N^6-32N^5+34N^4-20N^3+5N^2)
(6)
sum_(k=1)^(n)k^(13)=1/(105)(960N^7-2800N^6+4592N^5-4720N^4+2764N^3-691N^2)
(7)
sum_(k=1)^(n)k^(15)=1/3(48N^8-192N^7+448N^6-704N^5+718N^4-420N^3+105N^2)
(8)
sum_(k=1)^(n)k^(17)=1/(45)(1280N^9-6720N^8+21120N^7-46880N^6+72912N^5-74220N^4+43404N^3-10851N^2)
(9)

where N=(n^2+n)/2. While Faulhaber believed that analogous polynomials in N with alternating signs would continue to exist for all powers p, a rigorous proof was first published by Jacobi (1834; Knuth 1993).

Expressing such sums directly in terms of n for powers p=1, ..., 10 gives

sum_(k=1)^(n)k=1/2(n^2+n)
(10)
sum_(k=1)^(n)k^2=1/6(2n^3+3n^2+n)
(11)
sum_(k=1)^(n)k^3=1/4(n^4+2n^3+n^2)
(12)
sum_(k=1)^(n)k^4=1/(30)(6n^5+15n^4+10n^3-n)
(13)
sum_(k=1)^(n)k^5=1/(12)(2n^6+6n^5+5n^4-n^2)
(14)
sum_(k=1)^(n)k^6=1/(42)(6n^7+21n^6+21n^5-7n^3+n)
(15)
sum_(k=1)^(n)k^7=1/(24)(3n^8+12n^7+14n^6-7n^4+2n^2)
(16)
sum_(k=1)^(n)k^8=1/(90)(10n^9+45n^8+60n^7-42n^5+20n^3-3n)
(17)
sum_(k=1)^(n)k^9=1/(20)(2n^(10)+10n^9+15n^8-14n^6+10n^4-3n^2)
(18)
sum_(k=1)^(n)k^(10)=1/(66)(6n^(11)+33n^(10)+55n^9-66n^7+66n^5-33n^3+5n).
(19)

While Faulhaber was not aware of (and did not discover) Bernoulli numbers or harmonic numbers, a general formula for the sum of k^p for k from 1 to n can be given in closed form by

sum_(k=1)^(n)k^p=H_(n,-p)
(20)
=1/(p+1)sum_(i=1)^(p+1)(-1)^(delta_(ip))(p+1; i)B_(p+1-i)n^i,
(21)

where H_(n,r) is a generalized harmonic number, delta_(ip) is the Kronecker delta, (n; i) is a binomial coefficient, and B_i is the ith Bernoulli number.

In his work, Faulhaber also considered and (correctly) claimed that the r-fold summation of 1^p, 2^p, ..., n^p is a polynomial in n(n+r) when p=1 3, 5, .... Additional details are given by Knuth (1993, 2001).

Any of these power sums might be termed a "Faulhaber sum."


See also

Harmonic Number, Power, Power Sum, Sum

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References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 106, 1996.Edwards, A. W. F. "A Quick Route to Sums of Powers." Amer. Math. Monthly 93, 451-455, 1986.Faulhaber, J. Academia Algebræ, Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden. Augspurg [sic], Germany: Johann Ulrich Schönigs, 1631.Jacobi, C. G. J. "De usu legitimo formulae summatoriae Maclaurinianae." J. reine angew. Math. 12, 263-272, 1834.Knuth, D. E. "Johann Faulhaber and Sums of Powers." Math. Comput. 61, 277-294, 1993.Knuth, D. E. Ch. 4 in Selected Papers on Discrete Mathematics. Cambridge, England: Cambridge University Press, 2001.Schneider, I. Johannes Faulhaber 1580-1635: Rechenmeister in einer Welt des Umbruchs. Basel, Switzerland: Birkhäuser, 1993.

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Faulhaber's Formula

Cite this as:

Weisstein, Eric W. "Faulhaber's Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FaulhabersFormula.html

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