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A354042
Triangle read by rows. The Faulhaber numbers. F(0, k) = 1 and otherwise F(n, k) = (n + 1)!*(-1)^(k+1)*Sum_{j=0..floor((k-1)/2)} C(2*k-2*j, k+1)*C(2*n+1, 2*j+1) * Bernoulli(2*n-2*j) / (k - j).
3
1, 0, 1, 0, -1, 2, 0, 4, -8, 6, 0, -36, 72, -60, 24, 0, 600, -1200, 1020, -480, 120, 0, -16584, 33168, -28320, 13776, -4200, 720, 0, 705600, -1411200, 1206240, -591360, 188160, -40320, 5040, 0, -43751232, 87502464, -74813760, 36747648, -11813760, 2661120, -423360, 40320
OFFSET
0,6
COMMENTS
I. Gessel and X. Viennot call the rational numbers F(n, k)/(n + 1)! 'Faulhaber numbers'. However, for our purposes it is more convenient to define the integers F(n, k). For the Faulhaber polynomials see A335951/A335952.
Let S(r, m) = Sum_{k=0..m} k^r, with 0^0 = 1 and S(0, m) = m + 1. Faulhaber's theorem (the sums of powers formula) is:
S(2*n+1, m) = (1/(n+1)!)*(1/2)*Sum_{k=0..n} F(n, k)*(m*(m + 1))^(k + 1).
Gessel and Viennot give two combinatorial interpretations for the Faulhaber numbers, for this see A354043.
LINKS
I. M. Gessel and X. G. Viennot, Determinants, Paths, and Plane Partitions, 1989 preprint.
FORMULA
F(n,1) = (2*n +1)*Bernoulli(2*n)*(n+1)! for n >= 1.
F(n,2) = -(4*n+2)*Bernoulli(2*n)*(n+1)! for n >= 2.
F(n,3) = ((10*n+5)*Bernoulli(2*n) + binomial(2*n+1,3)*Bernoulli(2*n-2)/2)*(n+1)! for n >= 3.
EXAMPLE
Triangle starts:
0: 1
1: 0, 1
2: 0, -1, 2
3: 0, 4, -8, 6
4: 0, -36, 72, -60, 24
5: 0, 600, -1200, 1020, -480, 120
6: 0, -16584, 33168, -28320, 13776, -4200, 720
7: 0, 705600, -1411200, 1206240, -591360, 188160, -40320, 5040
8: 0, -43751232, 87502464, -74813760, 36747648, -11813760, 2661120, -423360, 40320
.
Let n = 4 and m = 3, then S(2*n + 1, m) = S(9, 3) = 20196. Faulhaber's formula gives this as (0*12 + (-36)*144 + 72*1728 + (-60)*20736 + 24*248832) / (2*120).
MAPLE
F := (n, k) -> ifelse(n = 0, 1, (n + 1)!*(-1)^(k + 1)*add(binomial(2*k - 2*j, k + 1)*binomial(2*n + 1, 2*j + 1)*bernoulli(2*n - 2*j) / (k - j), j = 0..(k - 1)/2)): for n from 0 to 8 do seq(F(n, k), k = 0..n) od;
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, May 17 2022
STATUS
approved