OFFSET
0,5
LINKS
FORMULA
a(n) = Sum_{k=0..floor(n/3)} 9^(n-3*k) * binomial(n-1-8*k/3,n-3*k).
From Vaclav Kotesovec, Feb 20 2024: (Start)
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)
MATHEMATICA
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 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(1/(1-x^3/(1-9*x)^(1/3)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Feb 06 2024
STATUS
approved