OFFSET
0,2
COMMENTS
Consider the sequence defined by a(0) = 1; thereafter a(n) = c*Sum_{k=1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.
Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255926. - Peter Bala, Mar 13 2015
In general, for c<>0 is e.g.f. = 1/(c+1-c*cosh(x)) (even coefficients). For c > 0 is a(n) ~ 2 * (2*n)! / (sqrt(2*c+1) * (arccosh((c+1)/c))^(2*n+1)). For c < 0 is a(n) ~ (-1)^n * (2*n)! / (sqrt(-2*c-1) * 2^(2*n) * arccos(sqrt((2*c + 1) / (2*c)))^(2*n+1)). - Vaclav Kotesovec, Mar 14 2015
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..200
FORMULA
E.g.f.: 1/(3*cosh(x)-2) (even coefficients). - Vaclav Kotesovec, Mar 14 2015
a(n) ~ (-1)^n * (2*n)! / (sqrt(5) * 2^(2*n) * (arccos(sqrt(5/6)))^(2*n+1)). - Vaclav Kotesovec, Mar 14 2015
MAPLE
f:=proc(n, k) option remember; local i;
if n=0 then 1
else k*add(binomial(2*n, 2*i)*f(n-i, k), i=1..floor(n)); fi; end;
g:=k->[seq(f(n, k), n=0..40)];
g(-3);
MATHEMATICA
nmax=20; Table[(CoefficientList[Series[1/(3*Cosh[x]-2), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n, 0, nmax}] (* Vaclav Kotesovec, Mar 14 2015 *)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
N. J. A. Sloane, Mar 28 2012
STATUS
approved