A338874 - OEIS

%I #38 Jan 20 2021 18:50:03

%S 1,1,1,1,2,1,1,6,6,1,1,1,36,24,1,1,30,180,1440,120,1,1,1,1080,11520,

%T 7200,720,1,1,42,9072,2419200,672000,1814400,5040,1,1,1,90720,2322432,

%U 60480000,435456000,12700800,40320,1,1,30,38880,232243200,207360000,548674560000,21337344000,270950400,362880,1

%N Array T(n, m) read by ascending antidiagonals: denominators of shifted Bernoulli numbers B(n, m) where m >= 0.

%H Stefano Spezia, <a href="/A338874/b338874.txt">First 30 antidiagonals of the array, flattened</a>

%H Takao Komatsu, <a href="https://www.researchgate.net/publication/344595540_SHIFTED_BERNOULLI_NUMBERS_AND_SHIFTED_FUBINI_NUMBERS">Shifted Bernoulli numbers and shifted Fubini numbers</a>, Linear and Nonlinear Analysis, Volume 6, Number 2, 2020, 245-263.

%F T(n, m) = denominator(B(n, m)).

%F B(n, m) = [x^n] n!*x^m/(exp(x) - E_m(x) + x^m), where E_m(x) = Sum_{n=0..m} x^n/n! (see Equation 2.1 in Komatsu).

%F B(n, m) = - Sum_{k=0..n-1} n!*B(k, m)/((n - k + m)!*k!) for n > 0 (see Lemma 2.1 in Komatsu).

%F B(n, m) = n!*Sum_{k=1..n} (-1)^k*Sum_{i_1+...+i_k=n; i_1,...,i_k>=1} Product_{j=1..k} 1/(i_j + m)! for n > 0 (see Theorem 2.2 in Komatsu).

%F B(n, m) = (-1)^n*n!*det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in 1/(m + 1)!, 1, 0, ..., 0 and whose first column consists in 1/(m + 1)!, 1/(m + 2)!, ..., 1/(m + n)! (see Theorem 2.3 in Komatsu).

%F B(1, m) = -1/(m + 1)! (see Theorem 2.4 in Komatsu).

%F B(n, m) = n!*Sum_{t_1+2*t_2+...+n*t_n=n} (t_1,...,t_n)!*(-1)^(t_1+…+t_n)*Product_{j=1..n} (1/(m + j)!)^t_j for n >= m >= 1 (see Theorem 2.7 in Komatsu).

%F (-1)^n/(n + m)! = det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in B(1, m), 1, 0, ..., 0 and whose first column consists in B(1, m), B(2, m)/2!, ..., B(n, m)/n! (see Theorem 2.8 in Komatsu).

%F Sum_{k=0..n} binomial(n, k)*B(k, m)*B(n-k, m) = - n!/(m^2*m!)*Sum_{l=0..n-1} ((m! - 1)/(m*m!))^(n-l-1)*(l*(m! - 1) + m)/l!*B(l, m) - (n - m)/m*B(n, m) for m > 0 (see Theorem 4.1 in Komatsu).

%e Array T(n, m):

%e n\m|   0         1         2         3         4 ...

%e ---+--------------------------------------------

%e 0  |   1         1         1         1         1 ...

%e 1  |   1         2         6        24       120 ...

%e 2  |   1         6        36      1440      7200 ...

%e 3  |   1         1       180     11520    672000 ...

%e 4  |   1        30      1080   2419200  60480000 ...

%e ...

%e Related table of shifted Bernoulli numbers B(n, m):

%e    1      1        1              1                1 ...

%e   -1   -1/2     -1/6          -1/24           -1/120 ...

%e    1    1/6    -1/36       -19/1440         -19/7200 ...

%e   -1      0    1/180      -53/11520      -709/672000 ...

%e    1  -1/30  11/1080  -3113/2419200  -28813/60480000 ...

%e   ...

%t B[n_, m_]:=n!Coefficient[Series[x^m/(Exp[x]-Sum[x^k/k!, {k, 0, m}]+x^m), {x, 0, n}], x, n]; Table[Denominator[B[n-m,m]],{n,0,9},{m,0,n}]//Flatten

%Y Cf. A000012 (1st column and 1st row), A000142 (2nd row), A027641, A027642 (2nd column), A141056, A164555, A176327, A226513 (high-order Fubini numbers), A338875, A338876.

%Y Cf. A338873 (numerators).

%K nonn,frac,tabl

%O 0,5

%A _Stefano Spezia_, Nov 13 2020