A338873 Array T(n, m) read by ascending antidiagonals: numerators of shifted Bernoulli numbers B(n, m) where m >= 0.

1, -1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 0, -1, -1, 1, -1, -1, 1, -19, -1, 1, 1, 0, 11, -53, -19, -1, 1, -1, 1, 43, -3113, -709, -713, -1, 1, 1, 0, -289, 349, -28813, -63367, -629, -1, 1, -1, -1, -313, 174947, -46721, -34877471, -351541, -1493, -1, 1, 1, 0, -581, 704101, -20744051, -2449743889, -176710589, -18054401, -36287, -1, 1

Offset

0,19

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) = numerator(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      -1      -1      -1      -1 ...
2  |   1       1      -1     -19     -19 ...
3  |  -1       0       1     -53    -709 ...
4  |   1      -1      11   -3113  -28813 ...
...
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[Numerator[B[n-m, m]], {n, 0, 10}, {m, 0, n}]//Flatten

Crossrefs

Cf. A000012 (1st row), A027641 (2nd column), A027642, A033999 (1st column), A141056, A164555, A176327, A226513 (high-order Fubini numbers), A338875, A338876.

Cf. A338874 (denominators).

Sequence in context: A317447 A202582 A292605 * A040365 A040366 A288145

Adjacent sequences: A338870 A338871 A338872 * A338874 A338875 A338876

Keyword

sign,frac,tabl

Author

Stefano Spezia, Nov 13 2020

Status

approved