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
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jun 04 2018
STATUS
approved