A210674 a(0)=1; thereafter a(n) = 3*Sum_{k=1..n} binomial(2n,2k)*a(n-k).

1, 3, 57, 2703, 239277, 34041603, 7103141697, 2043564786903, 775293596155317, 375019773885750603, 225270492555606688137, 164517775480287009524703, 143555042043378357951428157, 147502150365016885913874781203, 176273363579960990244526939543377, 242422256082395157286909073370272103

Offset: 0

Author

N. J. A. Sloane, Mar 28 2012

Keywords

• nonn
• easy

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 A255930. - Peter Bala, Mar 13 2015

In general, for c > 0 is a(n) ~ sqrt(Pi/(2*c+1)) * 2^(2*n+2) * n^(2*n+1/2) / (exp(2*n) * (log((c + 1 + sqrt(2*c+1)) / c))^(2*n+1)) = 2*(2*n)!/(sqrt(2*c+1)*(arccosh((c+1)/c))^(2*n+1)). - Vaclav Kotesovec, Mar 13 2015

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

• G. C. Greubel, Table of n, a(n) for n = 0..210

Crossrefs

Cf. A210676 (c=-3), A210657 (c=-2), A028296 (c=-1), A094088 (c=1), A210672 (c=2), A249939 (c=4).

Cf. A255930.

Programs

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/(4-3*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n,0,nmax}] (* Vaclav Kotesovec, Mar 14 2015 *)

Formula

a(n) ~ sqrt(Pi/7) * 2^(2*n+2) * n^(2*n+1/2) / (exp(2*n) * (log((4 + sqrt(7)) / 3))^(2*n+1)). - Vaclav Kotesovec, Mar 13 2015

E.g.f.: 1/(4-3*cosh(x)) (even coefficients). - Vaclav Kotesovec, Mar 14 2015