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 A335975 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). 2
 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, 1, 7, 41, 189, 622, 1215, 877, 1, 1, 8, 55, 311, 1357, 4117, 7107, 4140, 1, 1, 9, 71, 475, 2576, 10589, 29521, 44959, 21147, 1, 1, 10, 89, 687, 4447, 23031, 88909, 227290, 305091, 115975, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Seiichi Manyama, Antidiagonals n = 0..139, flattened FORMULA 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. T(n,k) = exp(-k) * Sum_{j>=0} (j + 1)^n * k^j / j!. EXAMPLE Square array begins:   1,   1,    1,     1,     1,      1,      1, ...   1,   2,    3,     4,     5,      6,      7, ...   1,   5,   11,    19,    29,     41,     55, ...   1,  15,   47,   103,   189,    311,    475, ...   1,  52,  227,   622,  1357,   2576,   4447, ...   1, 203, 1215,  4117, 10589,  23031,  44683, ...   1, 877, 7107, 29521, 88909, 220341, 478207, ... MATHEMATICA 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 *) CROSSREFS Columns k=0-4 give: A000012, A000110(n+1), A035009(n+1), A078940, A078945. Main diagonal gives A334240. Cf. A292860, A335977. Sequence in context: A094954 A083064 A204057 * A241578 A112338 A111672 Adjacent sequences:  A335972 A335973 A335974 * A335976 A335977 A335978 KEYWORD nonn,tabl AUTHOR Seiichi Manyama, Jul 03 2020 STATUS approved

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Last modified May 20 01:04 EDT 2022. Contains 353847 sequences. (Running on oeis4.)