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A369940
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Expansion of 1/(1 - x^3/(1-9*x)^(1/3)).
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2
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1, 0, 0, 1, 3, 18, 127, 951, 7416, 59329, 483147, 3986415, 33224338, 279121233, 2360156580, 20063973502, 171337660872, 1468794800925, 12633200032942, 108974515627170, 942420040015635, 8168578134973084, 70945593205544931, 617294050087428540
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OFFSET
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0,5
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LINKS
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FORMULA
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a(n) = Sum_{k=0..floor(n/3)} 9^(n-3*k) * binomial(n-1-8*k/3,n-3*k).
Recurrence (for n>19): (n-19)*(n-6)*(n-3)*a(n) = 9*(3*n^3 - 88*n^2 + 675*n - 1474)*a(n-1) - 9*(27*n^3 - 828*n^2 + 7067*n - 17838)*a(n-2) + 81*(n - 18)*(3*n - 25)*(3*n - 17)*a(n-3) + (n - 19)*(n-6)*(n-3)*a(n-9) - 18*(n^3 - 30*n^2 + 243*n - 566)*a(n-10) + 9*(n - 18)*(3*n - 25)*(3*n - 17)*a(n-11).
a(n) ~ (r-9)^(4/3) * r^(8/3) * r^n / (3*(r-8)), where r = 9.00000002323057264572143212814577340192663286000333917759... is the root of the equation (r-9)*r^8 = 1. (End)
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MATHEMATICA
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CoefficientList[Series[1/(1 - x^3/(1-9*x)^(1/3)), {x, 0, 25}], x] (* Vaclav Kotesovec, Feb 20 2024 *)
Join[{1, 0, 0, 1, 3, 18, 127, 951, 7416}, RecurrenceTable[{9 (-18 + n) (-25 + 3 n) (-17 + 3 n) a[-11 + n] - 18 (-566 + 243 n - 30 n^2 + n^3) a[-10 + n] + (-19 + n) (-6 + n) (-3 + n) a[-9 + n] + 81 (-18 + n) (-25 + 3 n) (-17 + 3 n) a[-3 + n] - 9 (-17838 + 7067 n - 828 n^2 + 27 n^3) a[-2 + n] + 9 (-1474 + 675 n - 88 n^2 + 3 n^3) a[-1 + n] - (-19 + n) (-6 + n) (-3 + n) a[n] == 0, a[9] == 59329, a[10] == 483147, a[11] == 3986415, a[12] == 33224338, a[13] == 279121233, a[14] == 2360156580, a[15] == 20063973502, a[16] == 171337660872, a[17] == 1468794800925, a[18] == 12633200032942, a[19] == 108974515627170}, a, {n, 9, 25}]] (* Vaclav Kotesovec, Feb 20 2024 *)
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PROG
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(PARI) my(N=30, x='x+O('x^N)); Vec(1/(1-x^3/(1-9*x)^(1/3)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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