A338875 Array T(n, m) read by ascending antidiagonals: numerators of shifted Fubini numbers F(n, m) where m >= 0.
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, 1, 1, 47293, 5081, 4157, 24967, 2687, 727, 1, 1, 545835, 674, 93881, 115567, 44027, 66247, 631, 1, 1, 7087261, 386237, 21209, 377909, 31627, 37728769, 354061, 4481, 1, 1
Offset: 0
Examples
Array T(n, m): n\m| 0 1 2 3 ... ---+-------------------------------- 0 | 1 1 1 1 ... 1 | 1 1 1 1 ... 2 | 3 5 5 29 ... 3 | 13 2 29 149 ... ... Related table of shifted Fubini numbers F(n, m): 1 1 1 1 ... 1 1/2 1/6 1/24 ... 3 5/6 5/36 29/1440 ... 13 2 29/180 149/11520 ... ...
Links
- 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
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 -
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; } T(n, m) = numerator(n!*matdet(tm(n, m))); \\ Michel Marcus, Dec 31 2020
Formula
T(n, m) = numerator(F(n, m)).
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(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(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(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(1, m) = 1/(m + 1)! (see Theorem 5.5 in Komatsu).
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).
(-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).
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).