A291652 -id:A291652 - cp's OEIS frontend

A291653 a(n) = [x^n] (1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...)))))))^n, a continued fraction.

1, 1, 3, 13, 55, 236, 1035, 4593, 20551, 92578, 419338, 1907951, 8713555, 39921038, 183396671, 844515563, 3896933367, 18014916576, 83415684654, 386807933378, 1796024496430, 8349190182990, 38854827380075, 180997895984903, 843906670596499, 3938005827167461, 18390418912425940

Offset: 0

Ilya Gutkovskiy, Aug 28 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Main diagonal of A291652.

Cf. A291274, A291651.

Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-x^i, 1, {i, 1, n}])^n, {x, 0, n}], {n, 0, 26}]

a(n) = A291652(n,n).

a(n) ~ c * d^n / sqrt(n), where d = 4.760595370947474723688065553003203505424287110594102605580439495640678... and c = 0.22756527349964754363249384886359862025065238... - Vaclav Kotesovec, Apr 08 2018

A291701 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 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...))))).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 1, -1, 0, 1, -4, 3, -2, 0, 0, 1, -5, 6, -4, 2, -1, 0, 1, -6, 10, -8, 6, -2, -1, 0, 1, -7, 15, -15, 13, -6, 1, -1, 0, 1, -8, 21, -26, 25, -16, 6, 0, -2, 0, 1, -9, 28, -42, 45, -36, 18, -3, 0, -2, 0, 1, -10, 36, -64, 77, -72

Offset: 0

Seiichi Manyama, Aug 30 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -2, -3, -4, ...
   0,  0,  1,  3,  6, ...
   0, -1, -2, -4, -8, ...
   0,  0,  2,  6, 13, ...

Columns k=0..1 give A000007, A291148.
Rows n=0..1 give A000012, A001489.
Main diagonal gives A291702.
Cf. A291652.

G.f. of column k: (1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...))))))^k, a continued fraction.

cp's OEIS Frontend

A291653 a(n) = [x^n] (1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...)))))))^n, a continued fraction.

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