OFFSET
0,12
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) = numerator(R(n, k)).
EXAMPLE
Triangle T(n, k) starts:
[0] 1;
[1] 1, 0;
[2] -1, 1, 0;
[3] 1, -1, 1, 0;
[4] -1, 11, -11, 1, 0;
[5] 1, -13, 11, -13, 1, 0;
[6] -1, 19, -151, 302, -19, 1, 0;
[7] 1, -5, 1191, -302, 397, -15, 1, 0;
[8] -1, 247, -477, 15619, -15619, 477, -247, 1, 0;
[9] 1, -251, 1826, -44117, 15619, -44117, 1826, -251, 1, 0;
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;
MAPLE
E1 := proc(n, k) combinat:-eulerian1(n, k) end:
Trow := proc(n, z) if n = 0 then return 1 fi;
seq(numer(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;
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[Numerator[T[n, k]], {n, 0, 8}, {k, 0, n}] // TableForm
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, Aug 15 2022
STATUS
approved