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

Author

Benedict W. J. Irwin, Feb 09 2016

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A000108, A003046.

Programs

Mathematica

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

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

Extensions

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