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A122068
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Expansion of x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3).
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3
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1, 3, 10, 35, 126, 462, 1715, 6419, 24157, 91238, 345401, 1309574, 4970070, 18874261, 71705865, 272491891, 1035680954, 3936821259, 14965658694, 56893879910, 216295686467, 822315097387, 3126323230541, 11885921055638
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = (1/2)*Sum_{k=0..2}(1 - 1/sqrt(7)*cot(2^k * alpha))* (2*sin(2^k * alpha))^(2n), where alpha := 2*Pi/7.
a(n) = binomial(2*n-1, n-1) + Sum_{k=1..n} (-1)^k*binomial(2*n, n+7*k). - Greg Dresden, Jan 28 2023
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MAPLE
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seq(coeff(series(x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3), x, n+1), x, n), n =1..30); # G. C. Greubel, Oct 03 2019
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MATHEMATICA
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M = {{2, 1, 0, 0, 0, 0}, {1, 2, 1, 0, 0, 0}, {0, 1, 2, 1, 0, 0}, {0, 0, 1, 2, 1, 0}, {0, 0, 0, 1, 2, 1}, {0, 0, 0, 0, 1, 2}}; v[1] = {1, 1, 1, 1, 1, 1}; v[n_]:= v[n] = M.v[n-1]; Table[v[n][[1]], {n, 30}]
Rest@CoefficientList[Series[x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3), {x, 0, 30}], x] (* G. C. Greubel, Oct 03 2019 *)
LinearRecurrence[{7, -14, 7}, {1, 3, 10}, 30] (* Harvey P. Dale, Mar 08 2020 *)
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PROG
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(Magma) I:=[1, 3, 10]; [n le 3 select I[n] else 7*(Self(n-1) -2*Self(n-2) + Self(n-3)): n in [1..30]]; // G. C. Greubel, Oct 03 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P(x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3)).list()
(GAP) a:=[1, 3, 10];; for n in [4..30] do a[n]:=7*(a[n-1]-2*a[n-2]+a[n-3]); od; a; # G. C. Greubel, Oct 03 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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