A209299 E.g.f.: 1 / Product_{n>=1} (cos(x^n/n) - sin(x^n/n)).

1, 1, 4, 16, 98, 650, 5492, 50468, 543252, 6375668, 83752144, 1191943168, 18563252968, 310499073352, 5598292885200, 107674197010960, 2208771882047120, 48025183073776016, 1105381958987588672, 26817991185065949440, 684717365565811694880, 18341702444087583851936

Offset

0,3

Comments

Compare to: 1/Product_{n>=1} (cosh(x^n/n) - sinh(x^n/n)) = 1/(1-x).

Limit (a(n)/n!)^(1/n) = 4/Pi; the radius of convergence of the e.g.f. is Pi/4.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Formula

a(n) ~ c * 2^(2*n+3/2) * n! / Pi^(n+1), where c = 1 / product_{n>=2} (cos((Pi/4)^n/n) - sin((Pi/4)^n/n)) = 2.516454534521990223577410114610797032290984895329... . - Vaclav Kotesovec, Nov 04 2014

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 16*x^3/3! + 98*x^4/4! + 650*x^5/5! +...
where A(x) = 1/((cos(x)-sin(x)) * (cos(x^2/2)-sin(x^2/2)) * (cos(x^3/3)-sin(x^3/3)) * (cos(x^4/4)-sin(x^4/4)) * (cos(x^5/5)-sin(x^5/5)) *...).

Mathematica

With[{nmax = 50}, CoefficientList[Series[1/Product[(Cos[x^n/n] - Sin[x^n/n]), {n, 1, 200}], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jan 03 2018 *)

Prog

(PARI) {a(n)=n!*polcoeff(1/prod(k=1, n, cos(x^k/k +x*O(x^n))-sin(x^k/k +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A209298.

Sequence in context: A130683 A111976 A236772 * A091040 A029985 A114023

Adjacent sequences: A209296 A209297 A209298 * A209300 A209301 A209302

Keyword

nonn

Author

Paul D. Hanna, Jan 17 2013

Status

approved