A369627 - OEIS

%I #19 Feb 19 2024 11:40:19

%S 1,0,1,3,19,132,991,7740,62020,505857,4180132,34889514,293518072,

%T 2485191753,21153817090,180865139538,1552289627872,13366436688402,

%U 115425148203235,999256943147094,8670047414816233,75375298322580081,656465004512563546

%N Expansion of 1/(1 - x^2/(1-9*x)^(1/3)).

%H Vaclav Kotesovec, <a href="/A369627/a369627.jpg">Graph - the asymptotic ratio (300000 terms)</a>

%F a(n) = Sum_{k=0..floor(n/2)} 9^(n-2*k) * binomial(n-1-5*k/3,n-2*k).

%F a(n) ~ (r-9)^(4/3) * r^(5/3) * r^n / (2*r-15), where r = 9.0000169349284790514638157821699098461789951085871459872133... = is the largest real root of the equation r^5*(r-9) = 1. - _Vaclav Kotesovec_, Feb 19 2024

%t CoefficientList[Series[1/(1 - x^2/(1-9*x)^(1/3)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 19 2024 *)

%t Flatten[{{1, 0, 1, 3, 19, 132}, RecurrenceTable[{9 (-12 + n) (-19 + 3 n) (-14 + 3 n) a[-8 + n] - 6 (-628 + 368 n - 63 n^2 + 3 n^3) a[-7 + n] + (-13 + n) (-4 + n) (-2 + n) a[-6 + n] + 81 (-12 + n) (-19 + 3 n) (-14 + 3 n) a[-3 + n] - 9 (-6960 + 3662 n - 585 n^2 + 27 n^3) a[-2 + n] + 3 (-14 + 3 n) (112 - 47 n + 3 n^2) a[-1 + n] - (-13 + n) (-4 + n) (-2 + n) a[n] == 0, a[6] == 991, a[7] == 7740, a[8] == 62020, a[9] == 505857, a[10] == 4180132, a[11] == 34889514, a[12] == 293518072, a[13] == 2485191753}, a, {n, 6, 20}]}] (* _Vaclav Kotesovec_, Feb 19 2024 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/(1-x^2/(1-9*x)^(1/3)))

%Y Cf. A362206, A369940.

%Y Cf. A104625.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Feb 06 2024