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%I #14 Feb 12 2019 12:05:51
%S 1,1,0,1,1,0,1,2,2,0,1,3,5,5,0,1,4,9,14,15,0,1,5,14,28,44,52,0,1,6,20,
%T 48,93,154,203,0,1,7,27,75,169,333,595,877,0,1,8,35,110,280,624,1289,
%U 2518,4140,0,1,9,44,154,435,1071,2442,5394,11591,21147,0,1,10,54,208,644,1728,4265,10188,24366,57672,115975,0
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of k-th power of continued fraction 1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))).
%C A(n,k) is the n-th term of the k-fold convolution of Bell numbers with themselves. - _Alois P. Heinz_, Feb 12 2019
%H Alois P. Heinz, <a href="/A292870/b292870.txt">Antidiagonals n = 0..140, flattened</a>
%F G.f. of column k: (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^k, a continued fraction.
%e G.f. of column k: A_k(x) = 1 + k*x + k*(k + 3)*x^2/2 + k*(k^2 + 9*k + 20)*x^3/6 + k*(k^3 + 18*k^2 + 107*k + 234)*x^4/24 + k*(k^4 + 30*k^3 + 335*k^2 + 1770*k + 4104)*x^5/120 + ...
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, ...
%e 0, 2, 5, 9, 14, 20, ...
%e 0, 5, 14, 28, 48, 75, ...
%e 0, 15, 44, 93, 169, 280, ...
%e 0, 52, 154, 333, 624, 1071, ...
%p A:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0,
%p `if`(k=1, add(A(n-j, k)*binomial(n-1, j-1), j=1..n),
%p (h-> add(A(j, h)*A(n-j, k-h), j=0..n))(iquo(k,2)))))
%p end:
%p seq(seq(A(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, May 31 2018
%t Table[Function[k, SeriesCoefficient[1/(1 - x + ContinuedFractionK[-i x^2, 1 - (i + 1) x, {i, 1, n}])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%Y Columns k=0-4 give A000007, A000110, A014322, A014323, A014325.
%Y Rows n=0-3 give A000012, A001477, A000096, A005586.
%Y Antidiagonal sums give A137551.
%Y Main diagonal gives A292871.
%Y Cf. A205574 (another version).
%K nonn,tabl
%O 0,8
%A _Ilya Gutkovskiy_, Sep 25 2017
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