A352646 - OEIS

%I #23 Mar 27 2022 02:59:14

%S 1,2,8,42,288,2410,24000,277186,3648512,53936082,885150720,

%T 15970846298,314273439744,6698574264122,153746319720448,

%U 3780677636321010,99163499845386240,2763481838977368994,81542013760903053312,2539717324111483027594

%N Expansion of e.g.f. 1/(1 - 2 * x * cos(x)).

%H Seiichi Manyama, <a href="/A352646/b352646.txt">Table of n, a(n) for n = 0..415</a>

%F a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (-1)^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).

%F a(n) ~ n! / ((1 - r * sqrt(4*r^2 - 1)) * r^n), where r = A196603 = 0.6100312844641759753709630735134103246737209791121692378637516075328... is the root of the equation 2*r*cos(r) = 1. - _Vaclav Kotesovec_, Mar 27 2022

%t With[{m = 19}, Range[0, m]! * CoefficientList[Series[1/(1 - 2*x*Cos[x]), {x, 0, m}], x]] (* _Amiram Eldar_, Mar 26 2022 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*x*cos(x))))

%o (PARI) a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\2, (-1)^k*(2*k+1)*binomial(n, 2*k+1)*a(n-2*k-1)));

%Y Cf. A352252, A352647.

%Y Cf. A352642, A352648.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 25 2022