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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 - ...)))))). 14
 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, 89, 1252, 3113, 132, 1, 1, 1, 7, 161, 5885, 104098, 106419, 429, 1, 1, 1, 8, 265, 20466, 1518897, 25511272, 7035649, 1430, 1, 1, 1, 9, 407, 57799, 12833546, 1558435125, 18649337311, 915028347, 4862, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS This is the transpose of the array in A090182. LINKS Seiichi Manyama, Antidiagonals n = 0..55, flattened FORMULA 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. EXAMPLE 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 + ... Square array begins:   1,   1,     1,       1,        1,         1,  ...   1,   1,     1,       1,        1,         1,  ...   1,   2,     3,       4,        5,         6,  ...   1,   5,    17,      43,       89,       161,  ...   1,  14,   171,    1252,     5885,     20466,  ...   1,  42,  3113,  104098,  1518897,  12833546,  ... MAPLE A:= proc(n, k) option remember; `if`(n=0, 1, add(       A(j, k)*A(n-j-1, k)*k^j, j=0..n-1))     end: seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 10 2017 MATHEMATICA 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 PROG (Python) from sympy.core.cache import cacheit @cacheit 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)) for n in range(13): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Aug 10 2017, after Maple code CROSSREFS Columns k=0-11 give: A000012, A000108, A015083, A015084, A015085, A015086, A015089,  A015091, A015092, A015093, A015095, A015096. Main diagonal gives A290777. Cf. A090182, A290789. Sequence in context: A305962 A144150 A124560 * A306245 A275043 A227061 Adjacent sequences:  A290756 A290757 A290758 * A290760 A290761 A290762 KEYWORD nonn,tabl AUTHOR Ilya Gutkovskiy, Aug 09 2017 STATUS approved

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