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A210676 a(0)=1; thereafter a(n) = -3*Sum_{k=1..n} binomial(2n,2k)*a(n-k). 7
1, -3, 51, -2163, 171231, -21785223, 4065116811, -1045879150683, 354837765112791, -153492920593758543, 82453488412268175171, -53850296379425229208803, 42020794900180632536559951, -38611325264740403135096141463, 41264215393801752999038147563131, -50749285521783354479522581233836523 (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 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

Table of n, a(n) for n=0..15.

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

Cf. A028296, A094088, A210657, A210672, A210674, A249939, A255926.

Sequence in context: A246693 A187666 A172434 * A105639 A003028 A069343

Adjacent sequences:  A210673 A210674 A210675 * A210677 A210678 A210679

KEYWORD

sign,easy

AUTHOR

N. J. A. Sloane, Mar 28 2012

STATUS

approved

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Last modified October 28 03:12 EDT 2021. Contains 348308 sequences. (Running on oeis4.)