%I #27 Jun 06 2024 13:08:24
%S 1,0,1,0,1,1,0,3,2,1,0,13,10,3,1,0,75,74,21,4,1,0,541,730,219,36,5,1,
%T 0,4683,9002,3045,484,55,6,1,0,47293,133210,52923,8676,905,78,7,1,0,
%U 545835,2299754,1103781,194404,19855,1518,105,8,1,0,7087261,45375130,26857659,5227236,544505,39390,2359,136,9,1
%N T(n, k) = F(n - k, k), where F(n, x) is the Fubini polynomial. Triangle read by rows, T(n, k) for 0 <= k <= n.
%C The array rows are recursively generated by applying the Akiyama-Tanigawa algorithm to the powers (see the Python implementation below). In this way the array becomes the image of A004248 under the AT-transformation when applied to the columns of A004248. This makes the array closely linked to A371761, which is generated in the same way, but applied to the rows of A004248. - _Peter Luschny_, Apr 27 2024
%F T(n, k) = (n - k)! * [x^(n - k)] (1 / (1 + k * (1 - exp(x)))).
%F T(2*n, n) = A094420(n).
%e Triangle starts:
%e [0] 1;
%e [1] 0, 1;
%e [2] 0, 1, 1;
%e [3] 0, 3, 2, 1;
%e [4] 0, 13, 10, 3, 1;
%e [5] 0, 75, 74, 21, 4, 1;
%e [6] 0, 541, 730, 219, 36, 5, 1;
%e [7] 0, 4683, 9002, 3045, 484, 55, 6, 1;
%e [8] 0, 47293, 133210, 52923, 8676, 905, 78, 7, 1;
%e [9] 0, 545835, 2299754, 1103781, 194404, 19855, 1518, 105, 8, 1;
%e .
%e Seen as an array A(n, k) = T(n + k, n):
%e [0] [1, 0, 0, 0, 0, 0, 0, ... A000007
%e [1] [1, 1, 3, 13, 75, 541, 4683, ... A000670
%e [2] [1, 2, 10, 74, 730, 9002, 133210, ... A004123
%e [3] [1, 3, 21, 219, 3045, 52923, 1103781, ... A032033
%e [4] [1, 4, 36, 484, 8676, 194404, 5227236, ... A094417
%e [5] [1, 5, 55, 905, 19855, 544505, 17919055, ... A094418
%e [6] [1, 6, 78, 1518, 39390, 1277646, 49729758, ... A094419
%e [7] [1, 7, 105, 2359, 70665, 2646007, 118893705, ... A238464
%p F := proc(n) option remember; if n = 0 then return 1 fi:
%p expand(add(binomial(n, k)*F(n - k)*x, k = 1..n)) end:
%p seq(seq(subs(x = k, F(n - k)), k = 0..n), n = 0..10);
%t F[n_] := F[n] = If[n == 0, 1,
%t Expand[Sum[Binomial[n, k]*F[n - k]*x, {k, 1, n}]]];
%t Table[Table[F[n - k] /. x -> k, {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jun 06 2024, after _Peter Luschny_ *)
%o (SageMath) # Computes the triangle.
%o @cached_function
%o def F(n):
%o R.<x> = PolynomialRing(ZZ)
%o if n == 0: return R(1)
%o return R(sum(binomial(n, k)*F(n - k)*x for k in (1..n)))
%o def Fval(n): return [F(n - k).substitute(x = k) for k in (0..n)]
%o for n in range(10): print(Fval(n))
%o (SageMath) # Computes the square array using the Akiyama-Tanigawa algorithm.
%o def ATFubini(n, len):
%o A = [0] * len
%o R = [0] * len
%o for k in range(len):
%o R[k] = (n + 1)**k # Chancing this to R[k] = k**n generates A371761.
%o for j in range(k, 0, -1):
%o R[j - 1] = j * (R[j] - R[j - 1])
%o A[k] = R[0]
%o return A
%o for n in range(8): print([n], ATFubini(n, 7)) # _Peter Luschny_, Apr 27 2024
%Y Variant of the array is A094416 (which has column 0 and row 0 missing).
%Y The coefficients of the Fubini polynomials are A131689.
%Y Cf. A094420 (main diagonal of array), A372346 (row sums), A004248, A371761.
%K nonn,tabl
%O 0,8
%A _Peter Luschny_, May 21 2021