This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A369940 #15 Feb 21 2024 03:13:27
%S A369940 1,0,0,1,3,18,127,951,7416,59329,483147,3986415,33224338,279121233,
%T A369940 2360156580,20063973502,171337660872,1468794800925,12633200032942,
%U A369940 108974515627170,942420040015635,8168578134973084,70945593205544931,617294050087428540
%N A369940 Expansion of 1/(1 - x^3/(1-9*x)^(1/3)).
%H A369940 Vaclav Kotesovec, <a href="/A369940/a369940.jpg">The asymptotic ratio (1000000 terms) - graph and Richardson extrapolation</a>
%F A369940 a(n) = Sum_{k=0..floor(n/3)} 9^(n-3*k) * binomial(n-1-8*k/3,n-3*k).
%F A369940 From _Vaclav Kotesovec_, Feb 20 2024: (Start)
%F A369940 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).
%F A369940 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)
%t A369940 CoefficientList[Series[1/(1 - x^3/(1-9*x)^(1/3)), {x, 0, 25}], x] (* _Vaclav Kotesovec_, Feb 20 2024 *)
%t A369940 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 *)
%o A369940 (PARI) my(N=30, x='x+O('x^N)); Vec(1/(1-x^3/(1-9*x)^(1/3)))
%Y A369940 Cf. A362206, A369627.
%K A369940 nonn
%O A369940 0,5
%A A369940 _Seiichi Manyama_, Feb 06 2024