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

Author

Ilya Gutkovskiy, Jun 04 2018

Keywords

• nonn

Comments

Invert transform of reduced tangent numbers (A002105).

Links

• N. J. A. Sloane, Transforms
• Index entries for sequences related to Bernoulli numbers

Crossrefs

Cf. A000217, A002105, A305532.

Programs

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}]

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