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
Examples
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 ... ...
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.
Crossrefs
Programs
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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
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).