A369627 Expansion of 1/(1 - x^2/(1-9*x)^(1/3)).

1, 0, 1, 3, 19, 132, 991, 7740, 62020, 505857, 4180132, 34889514, 293518072, 2485191753, 21153817090, 180865139538, 1552289627872, 13366436688402, 115425148203235, 999256943147094, 8670047414816233, 75375298322580081, 656465004512563546

Offset

0,4

Links

Table of n, a(n) for n=0..22.

Vaclav Kotesovec, Graph - the asymptotic ratio (300000 terms)

Formula

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

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

Mathematica

CoefficientList[Series[1/(1 - x^2/(1-9*x)^(1/3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 19 2024 *)
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 *)

Prog

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

Crossrefs

Cf. A362206, A369940.

Cf. A104625.

Sequence in context: A226459 A256092 A370159 * A074567 A332621 A091346

Adjacent sequences: A369624 A369625 A369626 * A369628 A369629 A369630

Keyword

nonn

Author

Seiichi Manyama, Feb 06 2024

Status

approved