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A210674 a(0)=1; thereafter a(n) = 3*Sum_{k=1..n} binomial(2n,2k)*a(n-k). 7
1, 3, 57, 2703, 239277, 34041603, 7103141697, 2043564786903, 775293596155317, 375019773885750603, 225270492555606688137, 164517775480287009524703, 143555042043378357951428157, 147502150365016885913874781203, 176273363579960990244526939543377, 242422256082395157286909073370272103 (list; graph; refs; listen; history; text; internal format)
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 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

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

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 *)

CROSSREFS

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

Cf. A255930.

Sequence in context: A069992 A012196 A012090 * A281184 A012064 A012204

Adjacent sequences:  A210671 A210672 A210673 * A210675 A210676 A210677

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Mar 28 2012

STATUS

approved

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Last modified April 26 08:11 EDT 2019. Contains 322472 sequences. (Running on oeis4.)