A305533 Expansion of 1/(1 - x/(1 - 1*x/(1 - 3*x/(1 - 6*x/(1 - 10*x/(1 - ... - (k*(k + 1)/2)*x/(1 - ...))))))), a continued fraction.

1, 1, 2, 7, 47, 592, 12287, 374857, 15639302, 851542747, 58536120467, 4953497262712, 505784457870707, 61300510121162077, 8698776162350603222, 1428545280744850604767, 268795232754158224790687, 57445320930331531152213232, 13837791987711934467999437927

Offset

0,3

Comments

Invert transform of reduced tangent numbers (A002105).

Links

Table of n, a(n) for n=0..18.

N. J. A. Sloane, Transforms

Index entries for sequences related to Bernoulli numbers

Formula

a(n) ~ 2^(3*n + 2) * n^(2*n - 1/2) / (exp(2*n) * Pi^(2*n - 1/2)). - Vaclav Kotesovec, Jun 08 2019

Maple

T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)
else (((k - n - 1)*(k - n)) / 2) * T(n, k - 1) + T(n - 1, k) fi fi end:
a := n -> T(n, n): seq(a(n), n = 0..18);  # Peter Luschny, Oct 01 2023

Mathematica

nmax = 18; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[-k (k + 1)/2 x, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 18; CoefficientList[Series[1/(1 - Sum[PolyGamma[2 k - 1, 1/2]/(2^(k - 2) Pi^(2 k)) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[2^k (2^(2 k) - 1) Abs[BernoulliB[2 k]]/k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

Crossrefs

Cf. A000217, A002105, A305532.

Sequence in context: A330149 A117141 A349965 * A125813 A254439 A341214

Adjacent sequences: A305530 A305531 A305532 * A305534 A305535 A305536

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 04 2018

Status

approved