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A292861 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*(1 - exp(x))).

%I A292861 #45 Dec 23 2021 06:04:53
%S A292861 1,1,0,1,-1,0,1,-2,0,0,1,-3,2,1,0,1,-4,6,2,1,0,1,-5,12,-3,-6,-2,0,1,
%T A292861 -6,20,-20,-21,-14,-9,0,1,-7,30,-55,-20,24,26,-9,0,1,-8,42,-114,45,
%U A292861 172,195,178,50,0,1,-9,56,-203,246,370,108,-111,90,267,0,1,-10,72,-328,679,318,-1105,-2388,-3072,-2382,413,0
%N A292861 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*(1 - exp(x))).
%H A292861 Seiichi Manyama, <a href="/A292861/b292861.txt">Antidiagonals n = 0..139, flattened</a>
%F A292861 A(0,k) = 1 and A(n,k) = -k * Sum_{j=0..n-1} binomial(n-1,j) * A(j,k) for n > 0.
%F A292861 A(n,k) = Sum_{j=0..n} (-k)^j * Stirling2(n,j). - _Seiichi Manyama_, Jul 27 2019
%F A292861 A(n,k) = BellPolynomial(n, -k). - _Peter Luschny_, Dec 23 2021
%e A292861 Square array begins:
%e A292861    1,  1,   1,   1,   1,     1,     1, ...
%e A292861    0, -1,  -2,  -3,  -4,    -5,    -6, ...
%e A292861    0,  0,   2,   6,  12,    20,    30, ...
%e A292861    0,  1,   2,  -3, -20,   -55,  -114, ...
%e A292861    0,  1,  -6, -21, -20,    45,   246, ...
%e A292861    0, -2, -14,  24, 172,   370,   318, ...
%e A292861    0, -9,  26, 195, 108, -1105, -4074, ...
%p A292861 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A292861       -(1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
%p A292861     end:
%p A292861 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Sep 25 2017
%t A292861 A[n_, k_] := Sum[(-k)^j StirlingS2[n, j], {j, 0, n}];
%t A292861 Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 10 2021 *)
%t A292861 A292861[n_, k_] := BellB[k, k - n];
%t A292861 Table[A292861[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Peter Luschny_, Dec 23 2021 *)
%Y A292861 Columns k=0..4 give A000007, A000587, A213170, A309084, A309085.
%Y A292861 Rows n=0..1 give A000012, (-1)*A001477.
%Y A292861 Main diagonal gives A292866.
%Y A292861 Cf. A292860, A309386.
%K A292861 sign,tabl,look
%O A292861 0,8
%A A292861 _Seiichi Manyama_, Sep 25 2017