A338874 Array T(n, m) read by ascending antidiagonals: denominators of shifted Bernoulli numbers B(n, m) where m >= 0.

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, 7200, 720, 1, 1, 42, 9072, 2419200, 672000, 1814400, 5040, 1, 1, 1, 90720, 2322432, 60480000, 435456000, 12700800, 40320, 1, 1, 30, 38880, 232243200, 207360000, 548674560000, 21337344000, 270950400, 362880, 1

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

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

Cf. A338873 (numerators).

Sequence in context: A199063 A140956 A166919 * A338876 A260238 A283795

Adjacent sequences: A338871 A338872 A338873 * A338875 A338876 A338877

Keyword

nonn,frac,tabl

Author

Stefano Spezia, Nov 13 2020

Status

approved