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A210676
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a(0)=1; thereafter a(n) = -3*Sum_{k=1..n} binomial(2n,2k)*a(n-k).
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9
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1, -3, 51, -2163, 171231, -21785223, 4065116811, -1045879150683, 354837765112791, -153492920593758543, 82453488412268175171, -53850296379425229208803, 42020794900180632536559951, -38611325264740403135096141463, 41264215393801752999038147563131, -50749285521783354479522581233836523
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OFFSET
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0,2
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COMMENTS
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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.
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
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LINKS
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FORMULA
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a(n) ~ (-1)^n * (2*n)! / (sqrt(5) * 2^(2*n) * (arccos(sqrt(5/6)))^(2*n+1)). - Vaclav Kotesovec, Mar 14 2015
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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