Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the 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