A305533 - OEIS

%I #11 Oct 01 2023 12:15:33

%S 1,1,2,7,47,592,12287,374857,15639302,851542747,58536120467,

%T 4953497262712,505784457870707,61300510121162077,8698776162350603222,

%U 1428545280744850604767,268795232754158224790687,57445320930331531152213232,13837791987711934467999437927

%N 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.

%C Invert transform of reduced tangent numbers (A002105).

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>

%F a(n) ~ 2^(3*n + 2) * n^(2*n - 1/2) / (exp(2*n) * Pi^(2*n - 1/2)). - _Vaclav Kotesovec_, Jun 08 2019

%p T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)

%p else (((k - n - 1)*(k - n)) / 2) * T(n, k - 1) + T(n - 1, k) fi fi end:

%p a := n -> T(n, n): seq(a(n), n = 0..18);  # _Peter Luschny_, Oct 01 2023

%t nmax = 18; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[-k (k + 1)/2 x, 1, {k, 1, nmax}])), {x, 0, nmax}], x]

%t 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]

%t 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}]

%Y Cf. A000217, A002105, A305532.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jun 04 2018