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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 (list; table; graph; refs; listen; history; text; internal format)
 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; CROSSREFS Cf. A335951/A335952, A000367/A002445, A354043, A263445. Sequence in context: A117902 A021087 A120558 * A325416 A120554 A120710 Adjacent sequences:  A354039 A354040 A354041 * A354043 A354044 A354045 KEYWORD sign,tabl AUTHOR Peter Luschny, May 17 2022 STATUS approved

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Last modified August 16 21:38 EDT 2022. Contains 356169 sequences. (Running on oeis4.)