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%I #15 Dec 10 2023 11:10:43
%S 1,1,0,-1,1,0,1,-1,1,0,-1,11,-11,1,0,1,-13,11,-13,1,0,-1,19,-151,302,
%T -19,1,0,1,-5,1191,-302,397,-15,1,0,-1,247,-477,15619,-15619,477,-247,
%U 1,0,1,-251,1826,-44117,15619,-44117,1826,-251,1,0
%N Triangle read by rows. T(n, k) = numerator(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.
%F 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.
%F Bernoulli(n) = Sum_{k=0..n} R(n, k), where Bernoulli(1) = 1/2.
%F T(n, k) = numerator(R(n, k)).
%e Triangle T(n, k) starts:
%e [0] 1;
%e [1] 1, 0;
%e [2] -1, 1, 0;
%e [3] 1, -1, 1, 0;
%e [4] -1, 11, -11, 1, 0;
%e [5] 1, -13, 11, -13, 1, 0;
%e [6] -1, 19, -151, 302, -19, 1, 0;
%e [7] 1, -5, 1191, -302, 397, -15, 1, 0;
%e [8] -1, 247, -477, 15619, -15619, 477, -247, 1, 0;
%e [9] 1, -251, 1826, -44117, 15619, -44117, 1826, -251, 1, 0;
%e The Bernoulli numbers (with B(1) = 1/2) are the row sums of the fractions.
%e [0] 1 = 1;
%e [1] + 1/2 = 1/2;
%e [2] - 1/6 + 1/3 = 1/6;
%e [3] + 1/12 - 1/3 + 1/4 = 0;
%e [4] - 1/20 + 11/30 - 11/20 + 1/5 = -1/30;
%e [5] + 1/30 - 13/30 + 11/10 - 13/15 + 1/6 = 0;
%e [6] - 1/42 + 19/35 - 151/70 + 302/105 - 19/14 + 1/7 = 1/42;
%p E1 := proc(n, k) combinat:-eulerian1(n, k) end:
%p Trow := proc(n, z) if n = 0 then return 1 fi;
%p seq(numer(int(E1(n, k)*z^(k + 1)*(z - 1)^(n - k - 1), z=0..1)), k=0..n) end:
%p for n from 0 to 9 do Trow(n, z) od;
%t Unprotect[Power]; Power[0, 0] = 1;
%t E1[n_, k_] /; n == k = 0^k; E1[n_, k_] /; k < 0 || k > n = 0;
%t E1[n_, k_] := E1[n, k] = (k + 1)*E1[n - 1, k] + (n - k)*E1[n - 1, k - 1];
%t T[n_, k_] /; n == k = 0^k;
%t T[n_, k_] := (-1)^(k - n + 1)*E1[n, k]*Gamma[k + 2]*Gamma[n - k]/Gamma[n + 2];
%t Table[Numerator[T[n, k]], {n, 0, 8}, {k, 0, n}] // TableForm
%Y Cf. A356601 (denominator), A173018, A278075, A356545, A356547.
%K sign,tabl,frac
%O 0,12
%A _Peter Luschny_, Aug 15 2022