OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..500
FORMULA
G.f. A=A(x) satisfies: A = 1 - 3*x*A + 3*x*A^2 + x*A^3.
G.f.: 1 + Series_Reversion( x/((1+x)*(1+5*x+x^2)) ).
G.f.: (1/x)*Series_Reversion( x*(1-3*x+sqrt((1-3*x)*(1-7*x)))/2/(1-3*x) ).
For n>0: a(n) = 1/n*sum(j=0..n, C(n,j) *sum(i=0..(n-1), C(j,i)*C(n-j,2*j-n-i-1) *6^(2*n-3*j+2*i+1))). - Vladimir Kruchinin, Dec 26 2010
a(n) ~ s^(3/2) / (3*sqrt(2*Pi*(1 + 3*s + 3*s^2)) * n^(3/2) * r^(n+1)), where s = 2*sin(Pi/6 + arctan(sqrt(7)/3)/3) - 1, r = 2*s/(9 - 12*sin(Pi/6 - 2*arctan(sqrt(7)/3)/3)). - Vaclav Kotesovec, Feb 18 2021
MATHEMATICA
T[n_, k_, p_]:= T[n, k, p] = If[n<k || k<0, 0, If[k==0, 1, If[k==n, 0, If[n<=2*k, T[n, n-k-1, p] + p*T[n-1, k, p], T[n, n-k, p] + T[n-1, k, p] ]]]];
Table[T[2*n, n, 3], {n, 0, 30}] (* G. C. Greubel, Feb 18 2021 *)
Join[{1}, Rest@CoefficientList[InverseSeries[Series[ x/((1+x)*(1+5*x+x^2)), {x, 0, 30}]], x]] (* G. C. Greubel, Feb 18 2021 *)
PROG
(PARI) {a(n)=polcoeff((serreverse(x*(1-3*x+sqrt((1-3*x)*(1-7*x)+x*O(x^n)))/2/(1-3*x))/x), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(1 + serreverse( x/((1+x)*(1+5*x+x^2 +x*O(x^n)))), n)}
for(n=0, 30, print1(a(n), ", "))
(Sage)
def S_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (x/((1+x)*(1+5*x+x^2))).reverse() ).list()
a=S_list(31); [1]+a[1:] # G. C. Greubel, Feb 18 2021
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
[1] cat Coefficients(R!( Reversion( x/((1+x)*(1+5*x+x^2)) ) )); // G. C. Greubel, Feb 18 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 26 2006
STATUS
approved