A356601 Triangle read by rows. T(n, k) = denominator(Integral_{z=0..1} Eulerian(n, k)*z^(k + 1)*(z - 1)^(n - k - 1) dz), where Eulerian(n, k) = A173018(n, k), for n >= 1, and T(0, 0) = 1.
1, 2, 1, 6, 3, 1, 12, 3, 4, 1, 20, 30, 20, 5, 1, 30, 30, 10, 15, 6, 1, 42, 35, 70, 105, 14, 7, 1, 56, 7, 280, 35, 56, 7, 8, 1, 72, 252, 56, 630, 504, 28, 72, 9, 1, 90, 180, 105, 630, 126, 420, 45, 45, 10, 1, 110, 495, 33, 1155, 1386, 1155, 165, 99, 110, 11, 1
Offset: 0
Examples
Triangle T(n, k) starts: [0] 1; [1] 2, 1; [2] 6, 3, 1; [3] 12, 3, 4, 1; [4] 20, 30, 20, 5, 1; [5] 30, 30, 10, 15, 6, 1; [6] 42, 35, 70, 105, 14, 7, 1; [7] 56, 7, 280, 35, 56, 7, 8, 1; [8] 72, 252, 56, 630, 504, 28, 72, 9, 1; [9] 90, 180, 105, 630, 126, 420, 45, 45, 10, 1; The Bernoulli numbers (with B(1) = 1/2) are the row sums of the fractions. [0] 1 = 1; [1] + 1/2 = 1/2; [2] - 1/6 + 1/3 = 1/6; [3] + 1/12 - 1/3 + 1/4 = 0; [4] - 1/20 + 11/30 - 11/20 + 1/5 = -1/30; [5] + 1/30 - 13/30 + 11/10 - 13/15 + 1/6 = 0; [6] - 1/42 + 19/35 - 151/70 + 302/105 - 19/14 + 1/7 = 1/42;
Links
- Grzegorz Rządkowski, Bernoulli numbers and solitons - revisited, Journal of Nonlinear Mathematical Physics, (2010) 17:1, 121-126.
Programs
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Maple
E1 := proc(n, k) combinat:-eulerian1(n, k) end: Trow := proc(n, z) if n = 0 then return 1 fi; seq(denom(int(E1(n, k)*z^(k + 1)*(z - 1)^(n - k - 1), z=0..1)), k=0..n) end: for n from 0 to 9 do Trow(n, z) od;
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Mathematica
Unprotect[Power]; Power[0, 0] = 1; E1[n_, k_] /; n == k = 0^k; E1[n_, k_] /; k < 0 || k > n = 0; E1[n_, k_] := E1[n, k] = (k + 1)*E1[n - 1, k] + (n - k)*E1[n - 1, k - 1]; T[n_, k_] /; n == k = 0^k; T[n_, k_] := (-1)^(k - n + 1)*E1[n, k]*Gamma[k + 2]*Gamma[n - k]/Gamma[n + 2]; Table[Denominator[T[n, k]], {n, 0, 8}, {k, 0, n}] // TableForm
Formula
R(n, k) = (-1)^(k - n + 1)*Eulerian(n, k)*Gamma(k + 2)*Gamma(n - k)/Gamma(n + 2) for 0 <= k < n, and T(n, n) = 0^n.
Bernoulli(n) = Sum_{k=0..n} R(n, k), where Bernoulli(1) = 1/2.
T(n, k) = denominator(R(n, k)).