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

0,4

Links

Table of n, a(n) for n=0..54.

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(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).

Example

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 ...
  ...

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

Prog

(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

Crossrefs

Cf. A000012 (n = 0 and n = 1), A000670 (m = 0), A226513 (high-order Fubini numbers), A232472, A232473, A232474, A257565, A338873, A338874.

Cf. A338876 (denominators).

Sequence in context: A094573 A055154 A373168 * A015112 A174690 A156869

Adjacent sequences: A338872 A338873 A338874 * A338876 A338877 A338878

Keyword

nonn,frac,tabl

Author

Stefano Spezia, Dec 25 2020

Status

approved