A356601 - OEIS

%I #19 Dec 10 2023 11:10:48

%S 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,

%T 1,56,7,280,35,56,7,8,1,72,252,56,630,504,28,72,9,1,90,180,105,630,

%U 126,420,45,45,10,1,110,495,33,1155,1386,1155,165,99,110,11,1

%N 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.

%H Grzegorz Rządkowski, <a href="https://doi.org/10.1142/S1402925110000635">Bernoulli numbers and solitons - revisited</a>, Journal of Nonlinear Mathematical Physics, (2010) 17:1, 121-126.

%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) = denominator(R(n, k)).

%e Triangle T(n, k) starts:

%e [0]  1;

%e [1]  2,   1;

%e [2]  6,   3,   1;

%e [3] 12,   3,   4,   1;

%e [4] 20,  30,  20,   5,   1;

%e [5] 30,  30,  10,  15,   6,   1;

%e [6] 42,  35,  70, 105,  14,   7,  1;

%e [7] 56,   7, 280,  35,  56,   7,  8,  1;

%e [8] 72, 252,  56, 630, 504,  28, 72,  9,  1;

%e [9] 90, 180, 105, 630, 126, 420, 45, 45, 10,  1;

%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(denom(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[Denominator[T[n, k]], {n, 0, 8}, {k, 0, n}] // TableForm

%Y Cf. A356602 (numerator), A173018, A278075, A356545, A356547.

%K nonn,tabl,frac

%O 0,2

%A _Peter Luschny_, Aug 15 2022