OFFSET
0,5
LINKS
Stefano Spezia, First 30 antidiagonals of the array, flattened
Takao Komatsu, Shifted Bernoulli numbers and shifted Fubini numbers, Linear and Nonlinear Analysis, Volume 6, Number 2, 2020, 245-263.
FORMULA
T(n, m) = denominator(B(n, m)).
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).
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).
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).
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).
B(1, m) = -1/(m + 1)! (see Theorem 2.4 in Komatsu).
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).
(-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).
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).
EXAMPLE
Array T(n, m):
n\m| 0 1 2 3 4 ...
---+--------------------------------------------
0 | 1 1 1 1 1 ...
1 | 1 2 6 24 120 ...
2 | 1 6 36 1440 7200 ...
3 | 1 1 180 11520 672000 ...
4 | 1 30 1080 2419200 60480000 ...
...
Related table of shifted Bernoulli numbers B(n, m):
1 1 1 1 1 ...
-1 -1/2 -1/6 -1/24 -1/120 ...
1 1/6 -1/36 -19/1440 -19/7200 ...
-1 0 1/180 -53/11520 -709/672000 ...
1 -1/30 11/1080 -3113/2419200 -28813/60480000 ...
...
MATHEMATICA
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
CROSSREFS
KEYWORD
AUTHOR
Stefano Spezia, Nov 13 2020
STATUS
approved