OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
invOGF satisfies n - (1+3*n)*a(n) - 2*n*(1+n)*a(n)^2 - 2*n^2*a(n)^3 = 0. [Is it true?]
Recurrence: (n+3)*(7*n-4)*a(n) = 3*(7*n^2+3*n+2)*a(n-1) + 2*(28*n^2+5*n-9)*a(n-2) + 4*(n-1)*(7*n+3)*a(n-3). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ sqrt(52+34*sqrt(2))*(2+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
EXAMPLE
{1}, {-2,-1,0}, {-1,0,1,2,3,0,1,2,-1,0,1}
MATHEMATICA
Length/@Flatten/@NestList[ # /. k_Integer:>Range[ -Abs[k+1], Abs[k-1]]&, {1}, 8]
Flatten[{1, RecurrenceTable[{(n+3)*(7*n-4)*a[n] == 3*(7*n^2+3*n+2)*a[n-1] + 2*(28*n^2+5*n-9)*a[n-2] + 4*(n-1)*(7*n+3)*a[n-3], a[1]==3, a[2]==11, a[3]==41}, a, {n, 20}]}] (* Vaclav Kotesovec, Oct 14 2012 *)
PROG
(Magma) I:=[1, 3, 11]; [n le 3 select I[n] else (3*(7*n^2 -11*n +6)*Self(n-1) + 2*(28*n^2 -51*n +14)*Self(n-2) + 4*(n-2)*(7*n-4)*Self(n-3))/((n+2)*(7*n-11)): n in [1..41]]; // G. C. Greubel, Nov 23 2022
(SageMath)
@CachedFunction
def a(n): # a = A084077
if (n<3): return (1, 3, 11)[n]
else: return (3*(7*n^2 +3*n +2)*a(n-1) + 2*(28*n^2 +5*n -9)*a(n-2) + 4*(n-1)*(7*n+3)*a(n-3))/((n+3)*(7*n-4))
[a(n) for n in range(31)] # G. C. Greubel, Nov 23 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Wouter Meeussen, May 11 2003
STATUS
approved