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A210676 a(0)=1; thereafter a(n) = -3*Sum_{k=1..n} binomial(2n,2k)*a(n-k).

%I A210676 #29 Jan 16 2022 08:19:01
%S A210676 1,-3,51,-2163,171231,-21785223,4065116811,-1045879150683,
%T A210676 354837765112791,-153492920593758543,82453488412268175171,
%U A210676 -53850296379425229208803,42020794900180632536559951,-38611325264740403135096141463,41264215393801752999038147563131,-50749285521783354479522581233836523
%N A210676 a(0)=1; thereafter a(n) = -3*Sum_{k=1..n} binomial(2n,2k)*a(n-k).
%C A210676 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.
%C A210676 Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255926. - _Peter Bala_, Mar 13 2015
%C A210676 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
%H A210676 Seiichi Manyama, <a href="/A210676/b210676.txt">Table of n, a(n) for n = 0..200</a>
%F A210676 E.g.f.: 1/(3*cosh(x)-2) (even coefficients). - _Vaclav Kotesovec_, Mar 14 2015
%F A210676 a(n) ~ (-1)^n * (2*n)! / (sqrt(5) * 2^(2*n) * (arccos(sqrt(5/6)))^(2*n+1)). - _Vaclav Kotesovec_, Mar 14 2015
%p A210676 f:=proc(n,k) option remember;  local i;
%p A210676 if n=0 then 1
%p A210676 else k*add(binomial(2*n,2*i)*f(n-i,k),i=1..floor(n)); fi; end;
%p A210676 g:=k->[seq(f(n,k),n=0..40)];
%p A210676 g(-3);
%t A210676 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 *)
%Y A210676 Cf. A028296, A094088, A210657, A210672, A210674, A249939, A255926.
%K A210676 sign,easy
%O A210676 0,2
%A A210676 _N. J. A. Sloane_, Mar 28 2012