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%I #28 Jan 01 2021 11:55:10
%S 1,1,1,3,1,1,13,5,1,1,75,2,5,1,1,541,191,29,29,1,1,4683,76,263,149,7,
%T 1,1,47293,5081,4157,24967,2687,727,1,1,545835,674,93881,115567,44027,
%U 66247,631,1,1,7087261,386237,21209,377909,31627,37728769,354061,4481,1,1
%N Array T(n, m) read by ascending antidiagonals: numerators of shifted Fubini numbers F(n, m) where m >= 0.
%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(F(n, m)).
%F F(n, m) = 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)^(n-1)/(m + n)! (see Proposition 5.1 in Komatsu).
%F F(n, m) = n!*Sum_{k=0..n-1} F(k, m)/((n - k + m)!*k!) for n > 0 and m >= 0 with F(0, m) = 1 (see Lemma 5.2).
%F F(n, m) = [x^n] n!*x^m/(x^m - exp(x) + E_m(x)), where E_m(x) = Sum_{n=0..m} x^n/n! (see Theorem 5.3 in Komatsu).
%F F(n, m) = n!*Sum_{k=1..n} Sum_{i_1+...+i_k=n, i_1,...,i_k>=1} Product_{j=1..k} 1/(i_j + m)! for n > 0 and m >= 0 (see Theorem 5.4).
%F F(1, m) = 1/(m + 1)! (see Theorem 5.5 in Komatsu).
%F F(n, m) = n!*Sum_{t_1+2*t_2+...+n*t_n=n} (t_1,...,t_n)!*(-1)^(n-t_1-...-t_n)*Product_{j=1..n} (1/(m + j)!)^t_j for n >= m >= 1 (see Theorem 5.7 in Komatsu).
%F (-1)^(n-1)/(n + m)! = det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in F(1, m), 1, 0, ..., 0 and whose first column consists in F(1, m), F(2, m)/2!, ..., F(n, m)/n! for n > 0 (see Theorem 5.8 in Komatsu).
%F Sum_{k=0..n} binomial(n, k)*F(k, m)*F(n-k, m) = - n!/(m^2*m!)*Sum_{l=0..n-1} ((m! + 1)/(m*m!))^(n-l-1)*(l*(m! + 1) - m)/l!*F(l, m) - (n - m)/m*F(n, m) for m > 0 (see Theorem 5.11 in Komatsu).
%e Array T(n, m):
%e n\m| 0 1 2 3 ...
%e ---+--------------------------------
%e 0 | 1 1 1 1 ...
%e 1 | 1 1 1 1 ...
%e 2 | 3 5 5 29 ...
%e 3 | 13 2 29 149 ...
%e ...
%e Related table of shifted Fubini numbers F(n, m):
%e 1 1 1 1 ...
%e 1 1/2 1/6 1/24 ...
%e 3 5/6 5/36 29/1440 ...
%e 13 2 29/180 149/11520 ...
%e ...
%t F[n_,m_]:=n!Coefficient[Series[x^m/(x^m-Exp[x]+Sum[x^k/k!,{k,0,m}]),{x,0,n}],x,n]; Table[Numerator[F[n-m,m]],{n,0,9},{m,0,n}]//Flatten
%o (PARI) tm(n, m) = {my(m = matrix(n, n, i, j, if (i==1, if (j==1, 1/(m + 1)!, if (j==2, 1)), if (j==1, (-1)^(i+1)/(m + i)!)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
%o T(n, m) = numerator(n!*matdet(tm(n, m))); \\ _Michel Marcus_, Dec 31 2020
%Y Cf. A000012 (n = 0 and n = 1), A000670 (m = 0), A226513 (high-order Fubini numbers), A232472, A232473, A232474, A257565, A338873, A338874.
%Y Cf. A338876 (denominators).
%K nonn,frac,tabl
%O 0,4
%A _Stefano Spezia_, Dec 25 2020
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