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 A189233 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals upwards, where the e.g.f. of column k is exp(k*(e^x-1)). 12
 1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 15, 22, 12, 4, 1, 0, 52, 94, 57, 20, 5, 1, 0, 203, 454, 309, 116, 30, 6, 1, 0, 877, 2430, 1866, 756, 205, 42, 7, 1, 0, 4140, 14214, 12351, 5428, 1555, 330, 56, 8, 1, 0, 21147, 89918, 88563, 42356, 12880, 2850, 497, 72, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS A(n, 1) = A000110(n), A(n, -1) = A000587(n). A(n,k) is the n-th moment of a Poisson distribution with mean = k. - Geoffrey Critzer, Dec 23 2018 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..5150 E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419. Peter Luschny, Set partitions and Bell numbers FORMULA E.g.f. of column k: exp(k*(e^x-1)). A(n,k) = Sum_{i=0..n} Stirling2(n,i)*k^i. - Vladimir Kruchinin, Apr 12 2019 EXAMPLE Square array begins: A000012   1,    1,    1,    1,    1,     1,     1,     1, ... A001477   0,    1,    2,    3,    4,     5,     6,     7, ... A002378   0,    2,    6,   12,   20,    30,    42,    56, ... A033445   0,    5,   22,   57,  116,   205,   330,   497, ...           0,   15,   94,  309,  756,  1555,  2850,  4809, ...           0,   52,  454, 1866, 5428, 12880, 26682, 50134, ... MAPLE # Cf. also the Maple prog. of Alois P. Heinz in A144223 and A144180. expnums := proc(k, n) option remember; local j; `if`(n = 0, 1, (1+add(binomial(n-1, j-1)*expnums(k, n-j), j = 1..n-1))*k) end: A189233_array := (k, n) -> expnums(k, n): seq(print(seq(A189233_array(k, n), k = 0..7)), n = 0..5); A189233_egf := k -> exp(k*(exp(x)-1)); T := (n, k) -> n!*coeff(series(A189233_egf(k), x, n+1), x, n): seq(lprint(seq(T(n, k), k = 0..7)), n = 0..5): # alternative Maple program: A:= proc(n, k) option remember; `if`(n=0, 1,       (1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)     end: seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Sep 25 2017 MATHEMATICA max = 9; Clear[col]; col[k_] := col[k] = CoefficientList[ Series[ Exp[k*(Exp[x]-1)], {x, 0, max}], x]*Range[0, max]!; a[0, _] = 1; a[n_?Positive, 0] = 0; a[n_, k_] := col[k][[n+1]]; Table[ a[n-k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 26 2013 *) Table[Table[BellB[n, k], {k, 0, 5}], {n, 0, 5}] // Grid  (* Geoffrey Critzer, Dec 23 2018 *) PROG (Maxima) A(n, k):=if k=0 and n=0 then 1 else if k=0 then 0 else  sum(stirling2(n, i)*k^i, i, 0, n); /* Vladimir Kruchinin, Apr 12 2019 */ CROSSREFS Columns: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263. Rows: A000012, A001477, A002378, A033445. Main diagonal gives A242817. Cf. A144150. Sequence in context: A067347 A120568 A321960 * A242153 A065066 A266291 Adjacent sequences:  A189230 A189231 A189232 * A189234 A189235 A189236 KEYWORD nonn,tabl AUTHOR Peter Luschny, Apr 18 2011 STATUS approved

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Last modified October 23 20:17 EDT 2019. Contains 328373 sequences. (Running on oeis4.)