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A290759 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - x/(1 - k*x/(1 - k^2*x/(1 - k^3*x/(1 - k^4*x/(1 - ...)))))).

%I A290759 #24 Jun 06 2020 12:59:13
%S A290759 1,1,1,1,1,1,1,1,2,1,1,1,3,5,1,1,1,4,17,14,1,1,1,5,43,171,42,1,1,1,6,
%T A290759 89,1252,3113,132,1,1,1,7,161,5885,104098,106419,429,1,1,1,8,265,
%U A290759 20466,1518897,25511272,7035649,1430,1,1,1,9,407,57799,12833546,1558435125,18649337311,915028347,4862,1
%N A290759 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - x/(1 - k*x/(1 - k^2*x/(1 - k^3*x/(1 - k^4*x/(1 - ...)))))).
%C A290759 This is the transpose of the array in A090182.
%H A290759 Seiichi Manyama, <a href="/A290759/b290759.txt">Antidiagonals n = 0..55, flattened</a>
%F A290759 G.f. of column k: 1/(1 - x/(1 - k*x/(1 - k^2*x/(1 - k^3*x/(1 - k^4*x/(1 - ...)))))), a continued fraction.
%e A290759 G.f. of column k: A_k(x) = 1 + x + (k + 1)*x^2 + (k^3 + k^2 + 2*k + 1)*x^3 + (k^6 + k^5 + 2*k^4 + 3*k^3 + 3*k^2 + 3*k + 1)*x^4 + ...
%e A290759 Square array begins:
%e A290759   1,   1,     1,       1,        1,         1,  ...
%e A290759   1,   1,     1,       1,        1,         1,  ...
%e A290759   1,   2,     3,       4,        5,         6,  ...
%e A290759   1,   5,    17,      43,       89,       161,  ...
%e A290759   1,  14,   171,    1252,     5885,     20466,  ...
%e A290759   1,  42,  3113,  104098,  1518897,  12833546,  ...
%p A290759 A:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A290759       A(j, k)*A(n-j-1, k)*k^j, j=0..n-1))
%p A290759     end:
%p A290759 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Aug 10 2017
%t A290759 Table[Function[k, SeriesCoefficient[1/(1 - x/(1 + ContinuedFractionK[-k^i x, 1, {i, 1, n}])), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%o A290759 (Python)
%o A290759 from sympy.core.cache import cacheit
%o A290759 @cacheit
%o A290759 def A(n, k): return 1 if n==0 else sum(A(j, k)*A(n - j - 1, k)*k**j for j in range(n))
%o A290759 for n in range(13): print([A(k, n - k) for k in range(n + 1)]) # _Indranil Ghosh_, Aug 10 2017, after Maple code
%Y A290759 Columns k=0-11 give: A000012, A000108, A015083, A015084, A015085, A015086, A015089,  A015091, A015092, A015093, A015095, A015096.
%Y A290759 Main diagonal gives A290777.
%Y A290759 Cf. A090182, A290789.
%K A290759 nonn,tabl
%O A290759 0,9
%A A290759 _Ilya Gutkovskiy_, Aug 09 2017