A268646 O.g.f.: 1/(1 - C(1)x/(1 - C(2)x/(1 - C(3)x/(1 - C(4)x/(1 - C(5)x/(1 - C(6)x/(1 -...))))))), a continued fraction, where C(n) are the Catalan numbers A000108.

1, 1, 3, 19, 277, 11081, 1383243, 569441699, 791393701997, 3770885471695081, 62402464265309818563, 3626978195203590614565619, 747715555141652980441024051237, 551447343768097359581617325419468841, 1465935896222119146302554598601016693710363

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..61

Formula

G.f.: 1/(1 - x/(1 - 2x/(1 - 5x/(1 - 14x/(1 - 42x/(1 -...)))))), by definition.

a(n) ~ c * A003046(n) ~ c * A^(3/2) * 2^(n^2+n-19/24) * exp(3*n/2-1/8) / (n^(3*n/2+15/8) * Pi^(n/2+1)), where A is the Glaisher-Kinkelin constant A074962 and c = 1/Product_{k>=1} (1 - 1/4^k) = 1/QPochhammer[1/4] = 1.452353642449597... - Vaclav Kotesovec, Aug 26 2017

Mathematica

Table[SeriesCoefficient[Series[1/(1+ContinuedFractionK[-CatalanNumber[k]*x, 1, {k, 1, 50}]), {x, 0, 50}], n], {n, 1, 50}]

Crossrefs

Cf. A000108, A003046.

Sequence in context: A003011 A338772 A231620 * A356512 A143597 A224681

Adjacent sequences: A268643 A268644 A268645 * A268647 A268648 A268649

Keyword

nonn,easy

Author

Benedict W. J. Irwin, Feb 09 2016

Extensions

a(0) = 1 added by Peter Bala, Apr 17 2017

Status

approved