A291261 -id:A291261 - cp's OEIS frontend

A291332 a(n) = [x^n] 1/(1 - x/(1 - 3^nx/(1 - 5^nx/(1 - 7^nx/(1 - 9^nx/(1 - ...)))))), a continued fraction.

1, 1, 10, 4159, 162045118, 1063421637466546, 1858323116289048481112500, 1253322341309506161980784960477550459, 445827827888374514639499681047571455105640696771958, 109534636154930845670316103395158313783593902542091687316468724140446

Offset: 0

Ilya Gutkovskiy, Aug 22 2017

nonn

Main diagonal of A291261.

Cf. A291547.

Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-(2 i - 1)^n x, 1, {i, 1, n}]), {x, 0, n}], {n, 0, 9}]

a(n) = A291261(n,n).

a(n) ~ c * ((2*n-1)!!)^n ~ c * 2^(n^2 + n/2) * n^(n^2) / exp(n^2 + 1/24), where c = 1/QPochhammer(exp(-1)) = 1.9824409074128737036856824655613120156828827... - Vaclav Kotesovec, Aug 26 2017, updated Jul 21 2018

A291260 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - 2^kx/(1 - 4^kx/(1 - 6^kx/(1 - 8^kx/(1 - 10^k*x/(1 - ...)))))).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 12, 5, 1, 8, 80, 120, 14, 1, 16, 576, 3904, 1680, 42, 1, 32, 4352, 152064, 354560, 30240, 132, 1, 64, 33792, 6492160, 99422208, 51733504, 665280, 429, 1, 128, 266240, 290488320, 31832735744, 130292416512, 11070525440, 17297280, 1430

Offset: 0

Ilya Gutkovskiy, Aug 21 2017

			Square array begins:
:  1,     1,        1,            1,               1, ...
:  1,     2,        4,            8,              16, ...
:  2,    12,       80,          576,            4352, ...
:  5,   120,     3904,       152064,         6492160, ...
: 14,  1680,   354560,     99422208,     31832735744, ...
: 42, 30240, 51733504, 130292416512, 390365719822336, ...

Columns k=0-2 give A000108, A001813, A002436.
Main diagonal gives A291331.
Cf. A000079 (row 1), A063481 (row 2), A290569, A291261.

Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-(2 i)^k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 8}, {n, 0, j}] // Flatten

G.f. of column k: 1/(1 - 2^k*x/(1 - 4^k*x/(1 - 6^k*x/(1 - 8^k*x/(1 - 10^k*x/(1 - ...)))))), a continued fraction.

cp's OEIS Frontend

A291332 a(n) = [x^n] 1/(1 - x/(1 - 3^nx/(1 - 5^nx/(1 - 7^nx/(1 - 9^nx/(1 - ...)))))), a continued fraction.

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A291260 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - 2^kx/(1 - 4^kx/(1 - 6^kx/(1 - 8^kx/(1 - 10^k*x/(1 - ...)))))).

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cp's OEIS Frontend

A291332 a(n) = [x^n] 1/(1 - x/(1 - 3^n*x/(1 - 5^n*x/(1 - 7^n*x/(1 - 9^n*x/(1 - ...)))))), a continued fraction.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A291260 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - 2^k*x/(1 - 4^k*x/(1 - 6^k*x/(1 - 8^k*x/(1 - 10^k*x/(1 - ...)))))).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A291332 a(n) = [x^n] 1/(1 - x/(1 - 3^nx/(1 - 5^nx/(1 - 7^nx/(1 - 9^nx/(1 - ...)))))), a continued fraction.

A291260 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - 2^kx/(1 - 4^kx/(1 - 6^kx/(1 - 8^kx/(1 - 10^k*x/(1 - ...)))))).