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A356602 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.

%I A356602 #15 Dec 10 2023 11:10:43
%S A356602 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 A356602 -19,1,0,1,-5,1191,-302,397,-15,1,0,-1,247,-477,15619,-15619,477,-247,
%U A356602 1,0,1,-251,1826,-44117,15619,-44117,1826,-251,1,0
%N A356602 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 A356602 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 A356602 Bernoulli(n) = Sum_{k=0..n} R(n, k), where Bernoulli(1) = 1/2.
%F A356602 T(n, k) = numerator(R(n, k)).
%e A356602 Triangle T(n, k) starts:
%e A356602 [0]  1;
%e A356602 [1]  1,    0;
%e A356602 [2] -1,    1,    0;
%e A356602 [3]  1,   -1,    1,      0;
%e A356602 [4] -1,   11,  -11,      1,      0;
%e A356602 [5]  1,  -13,   11,    -13,      1,      0;
%e A356602 [6] -1,   19, -151,    302,    -19,      1,    0;
%e A356602 [7]  1,   -5, 1191,   -302,    397,    -15,    1,    0;
%e A356602 [8] -1,  247, -477,  15619, -15619,    477, -247,    1,  0;
%e A356602 [9]  1, -251, 1826, -44117,  15619, -44117, 1826, -251,  1,  0;
%e A356602 The Bernoulli numbers (with B(1) = 1/2) are the row sums of the fractions.
%e A356602 [0]   1                                              =     1;
%e A356602 [1] + 1/2                                            =   1/2;
%e A356602 [2] - 1/6  +  1/3                                    =   1/6;
%e A356602 [3] + 1/12 -  1/3  +    1/4                          =     0;
%e A356602 [4] - 1/20 + 11/30 -  11/20 +   1/5                  = -1/30;
%e A356602 [5] + 1/30 - 13/30 +  11/10 -  13/15  +   1/6        =     0;
%e A356602 [6] - 1/42 + 19/35 - 151/70 + 302/105 - 19/14 + 1/7  =  1/42;
%p A356602 E1 := proc(n, k) combinat:-eulerian1(n, k) end:
%p A356602 Trow := proc(n, z) if n = 0 then return 1 fi;
%p A356602 seq(numer(int(E1(n, k)*z^(k + 1)*(z - 1)^(n - k - 1), z=0..1)), k=0..n) end:
%p A356602 for n from 0 to 9 do Trow(n, z) od;
%t A356602 Unprotect[Power]; Power[0, 0] = 1;
%t A356602 E1[n_, k_] /; n == k = 0^k; E1[n_, k_] /; k < 0 || k > n = 0;
%t A356602 E1[n_, k_] := E1[n, k] = (k + 1)*E1[n - 1, k] + (n - k)*E1[n - 1, k - 1];
%t A356602 T[n_, k_] /; n == k = 0^k;
%t A356602 T[n_, k_] := (-1)^(k - n + 1)*E1[n, k]*Gamma[k + 2]*Gamma[n - k]/Gamma[n + 2];
%t A356602 Table[Numerator[T[n, k]], {n, 0, 8}, {k, 0, n}] // TableForm
%Y A356602 Cf. A356601 (denominator), A173018, A278075, A356545, A356547.
%K A356602 sign,tabl,frac
%O A356602 0,12
%A A356602 _Peter Luschny_, Aug 15 2022