A305532 Expansion of 1/(1 - x/(1 - 1*2*x/(1 - 2*3*x/(1 - 3*4*x/(1 - 4*5*x/(1 - ...)))))), a continued fraction.

1, 1, 3, 21, 315, 8613, 372123, 23145957, 1951467291, 213852190437, 29523337936155, 5011258121042661, 1025542423300379931, 248988579422292953829, 70752815796279635539227, 23261468728483619098626789, 8760705555494801063319729435, 3747001028007419861036996070117

Offset

0,3

Comments

Invert transform of tangent numbers (A000182).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..243

N. J. A. Sloane, Transforms

Index entries for sequences related to Bernoulli numbers

Formula

a(n) ~ 2^(4*n + 1) * n^(2*n - 1/2) / (exp(2*n) * Pi^(2*n - 1/2)). - Vaclav Kotesovec, Jun 08 2019

Maple

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, max(1, y)*(b(x-1, y-1)+b(x-1, y+1))))
    end:
a:= n->  b(2*n, 0) :
seq(a(n), n=0..22);  # Alois P. Heinz, Jun 08 2018
# Alternative:
T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)
else (n - k)*(n - k + 1) * T(n, k - 1) + T(n - 1, k) fi fi end:
a := n -> T(n, n): seq(a(n), n = 0..17);  # Peter Luschny, Oct 01 2023

Mathematica

nmax = 17; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[-k (k + 1) x, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 17; CoefficientList[Series[1/(1 - Sum[2 PolyGamma[2 k - 1, 1/2]/Pi^(2 k) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[2^(2 k) (2^(2 k) - 1) Abs[BernoulliB[2 k]]/(2 k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]

Crossrefs

Cf. A000182, A002378, A303943, A305533.

Sequence in context: A208731 A158888 A331583 * A005329 A341471 A134528

Adjacent sequences: A305529 A305530 A305531 * A305533 A305534 A305535

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 04 2018

Status

approved