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%I #42 Jan 20 2021 18:49:50
%S 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,
%T -1,1,-1,1,43,-3113,-709,-713,-1,1,1,0,-289,349,-28813,-63367,-629,-1,
%U 1,-1,-1,-313,174947,-46721,-34877471,-351541,-1493,-1,1,1,0,-581,704101,-20744051,-2449743889,-176710589,-18054401,-36287,-1,1
%N Array T(n, m) read by ascending antidiagonals: numerators of shifted Bernoulli numbers B(n, m) where m >= 0.
%H Stefano Spezia, <a href="/A338873/b338873.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) = numerator(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 -1 -1 -1 -1 ...
%e 2 | 1 1 -1 -19 -19 ...
%e 3 | -1 0 1 -53 -709 ...
%e 4 | 1 -1 11 -3113 -28813 ...
%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[Numerator[B[n-m,m]],{n,0,10},{m,0,n}]//Flatten
%Y Cf. A000012 (1st row), A027641 (2nd column), A027642, A033999 (1st column), A141056, A164555, A176327, A226513 (high-order Fubini numbers), A338875, A338876.
%Y Cf. A338874 (denominators).
%K sign,frac,tabl
%O 0,19
%A _Stefano Spezia_, Nov 13 2020
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