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A290759
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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 - ...)))))).
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14
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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
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OFFSET
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0,9
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COMMENTS
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This is the transpose of the array in A090182.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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, ...
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MAPLE
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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:
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MATHEMATICA
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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
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PROG
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(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
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CROSSREFS
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Columns k=0-11 give: A000012, A000108, A015083, A015084, A015085, A015086, A015089, A015091, A015092, A015093, A015095, A015096.
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KEYWORD
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AUTHOR
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STATUS
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approved
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