login
Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1) + x).
4

%I #19 Jul 06 2020 10:41:00

%S 1,1,1,1,2,1,1,3,5,1,1,4,11,15,1,1,5,19,47,52,1,1,6,29,103,227,203,1,

%T 1,7,41,189,622,1215,877,1,1,8,55,311,1357,4117,7107,4140,1,1,9,71,

%U 475,2576,10589,29521,44959,21147,1,1,10,89,687,4447,23031,88909,227290,305091,115975,1

%N Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1) + x).

%H Seiichi Manyama, <a href="/A335975/b335975.txt">Antidiagonals n = 0..139, flattened</a>

%F T(0,k) = 1 and T(n,k) = T(n-1,k) + k * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

%F T(n,k) = exp(-k) * Sum_{j>=0} (j + 1)^n * k^j / j!.

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 4, 5, 6, 7, ...

%e 1, 5, 11, 19, 29, 41, 55, ...

%e 1, 15, 47, 103, 189, 311, 475, ...

%e 1, 52, 227, 622, 1357, 2576, 4447, ...

%e 1, 203, 1215, 4117, 10589, 23031, 44683, ...

%e 1, 877, 7107, 29521, 88909, 220341, 478207, ...

%t T[0, k_] := 1; T[n_, k_] := T[n - 1, k] + k * Sum[T[j, k] * Binomial[n - 1, j], {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Amiram Eldar_, Jul 03 2020 *)

%Y Columns k=0-4 give: A000012, A000110(n+1), A035009(n+1), A078940, A078945.

%Y Main diagonal gives A334240.

%Y Cf. A292860, A335977.

%K nonn,tabl

%O 0,5

%A _Seiichi Manyama_, Jul 03 2020