A367518 Numerators of even-numbered Maclaurin coefficients of sqrt(tan(x)/x).

1, 1, 19, 55, 11813, 2117, 64604977, 263101079, 1768132943, 9606907803497, 158812278992229461, 9112944418860287, 2117852079027536379043, 27841657661565660151, 909416652267282749299777, 26176589384334728915393123, 22901449589921151647801250738173, 514908297269179169530303586629

Offset: 0

Author

Robert Israel, Nov 21 2023

Keywords

• nonn
• frac

Comments

Numerators of Maclaurin coefficients of sqrt(tan(sqrt(x)))/x^(1/4).

Examples

			sqrt(tan(x)/x) = 1 + (1/6) * x^2 + (19/360) * x^4 + (55/3024) * x^6 + ...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..241

Crossrefs

Cf. A002430, A367519.

Programs

Maple

S:= series(sqrt(tan(x)/x), x, 41):
seq(numer(coeff(S,x,i)),i=0..40,2);

Maxima

b[n]:=if n=0 then 1 else sum(b[n+1-j]*bern(2*j)*((-4)^j-(-16)^j)*((3*j-3)/(2*n)-1)/(2*j)!,j,2,n+1)$ a(n):=num(b[n])$ makelist(a(n),n,0,30); /* Tani Akinari, Feb 26 2025 */

PARI

my(x='x+O('x^40), v=apply(numerator, Vec(sqrt(tan(x)/x)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Feb 26 2025

Formula

sqrt(tan(x)/x) = Sum_{k=0..oo} a(k)/A367519(k) * x^(2*k).

a(n) = numerator(b(n)), where b(n) = Sum_{j=2..n+1} b(n+1-j)*Bernoulli(2*j)*((-4)^j-(-16)^j)*((3*j-3)/(2*n)-1)/(2*j)!, with b(0)=1. - Tani Akinari, Feb 26 2025