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A291652 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/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...)))))).

%I A291652 #11 Feb 16 2025 08:33:50
%S A291652 1,1,0,1,1,0,1,2,1,0,1,3,3,2,0,1,4,6,6,3,0,1,5,10,13,11,5,0,1,6,15,24,
%T A291652 27,20,9,0,1,7,21,40,55,54,38,15,0,1,8,28,62,100,120,109,70,26,0,1,9,
%U A291652 36,91,168,236,258,216,129,45,0,1,10,45,128,266,426,540,544,423,238,78,0,1,11,55,174,402,721,1035,1205,1127,824,437,135,0
%N A291652 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/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...)))))).
%H A291652 Seiichi Manyama, <a href="/A291652/b291652.txt">Antidiagonals n = 0..139, flattened</a>
%H A291652 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%F A291652 G.f. of column k: (1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...)))))))^k, a continued fraction.
%e A291652 G.f. of column k: A_k(x) = 1 + k*x + k*(k + 1)*x^2/2 +  k*(k^2 + 3*k + 8)*x^3/6 + k*(k^3 + 6*k^2 + 35*k + 30)*x^4/24 + ...
%e A291652 Square array begins:
%e A291652 1,  1,   1,   1,    1,    1,  ...
%e A291652 0,  1,   2,   3,    4,    5,  ...
%e A291652 0,  1,   3,   6,   10,   15,  ...
%e A291652 0,  2,   6,  13,   24,   40,  ...
%e A291652 0,  3,  11,  27,   55,  100,  ...
%e A291652 0,  5,  20,  54,  120,  236,  ...
%t A291652 Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-x^i, 1, {i, 1, n}])^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
%t A291652 Table[Function[k, SeriesCoefficient[((Sum[(-1)^i x^(i (i + 1))/Product[(1 - x^m), {m, 1, i}], {i, 0, n}])/(Sum[(-1)^i x^(i^2)/Product[(1 - x^m), {m, 1, i}], {i, 0, n}]))^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
%Y A291652 Columns k=0..1 give A000007, A005169.
%Y A291652 Rows n=0..3 give A000012, A001477, A000217, A003600 (with a(0)=0).
%Y A291652 Main diagonal gives A291653.
%Y A291652 Cf. A286509, A286933.
%K A291652 nonn,tabl
%O A291652 0,8
%A A291652 _Ilya Gutkovskiy_, Aug 28 2017