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 A293024 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(exp(x) - Sum_{i=0..k} x^i/i!). 7
 1, 1, 1, 1, 0, 2, 1, 0, 1, 5, 1, 0, 0, 1, 15, 1, 0, 0, 1, 4, 52, 1, 0, 0, 0, 1, 11, 203, 1, 0, 0, 0, 1, 1, 41, 877, 1, 0, 0, 0, 0, 1, 11, 162, 4140, 1, 0, 0, 0, 0, 1, 1, 36, 715, 21147, 1, 0, 0, 0, 0, 0, 1, 1, 92, 3425, 115975, 1, 0, 0, 0, 0, 0, 1, 1, 36, 491, 17722, 678570 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS A(n,k) is the number of set partitions of [n] into blocks of size > k. LINKS Seiichi Manyama, Antidiagonals n = 0..139, flattened E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73. FORMULA E.g.f. of column k: Product_{i>k} exp(x^i/i!). A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = Sum_{i=k..n-1} binomial(n-1,i)*A(n-1-i,k) for n > k. EXAMPLE Square array begins:     1,  1, 1, 1, 1, ...     1,  0, 0, 0, 0, ...     2,  1, 0, 0, 0, ...     5,  1, 1, 0, 0, ...    15,  4, 1, 1, 0, ...    52, 11, 1, 1, 1, ... MAPLE A:= proc(n, k) option remember; `if`(n=0, 1, add(       A(n-j, k)*binomial(n-1, j-1), j=1+k..n))     end: seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Sep 28 2017 MATHEMATICA A[0, _] = 1; A[n_, k_] /; 0 <= k <= n := A[n, k] = Sum[A[n-j, k] Binomial[n-1, j-1], {j, k+1, n}]; A[_, _] = 0; Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2019 *) PROG (Ruby) def ncr(n, r)   return 1 if r == 0   (n - r + 1..n).inject(:*) / (1..r).inject(:*) end def A(k, n)   ary = [1]   (1..n).each{|i| ary << (k..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[i - 1 - j]}}   ary end def A293024(n)   a = []   (0..n).each{|i| a << A(i, n - i)}   ary = []   (0..n).each{|i|     (0..i).each{|j|       ary << a[i - j][j]     }   }   ary end p A293024(20) CROSSREFS Columns k=0..5 give A000110, A000296, A006505, A057837, A057814, A293025. Rows n=0..1 give A000012, A000007. Main diagonal gives A000007. Cf. A282988 (as triangle). Sequence in context: A266972 A266493 A075374 * A292948 A210872 A292973 Adjacent sequences:  A293021 A293022 A293023 * A293025 A293026 A293027 KEYWORD nonn,tabl AUTHOR Seiichi Manyama, Sep 28 2017 STATUS approved

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Last modified August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)