OFFSET
0,3
FORMULA
G.f.: 1 + Series_Reversion(x - 3*x^2 - 6*x^3 - 4*x^4 - x^5).
G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(6*n) * Product_{k=1..n} (1 - 1/A(x)^(3*k-2)).
G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(n*(3*n-1)/2+6*n)) * Product_{k=1..n} (A(x)^(3*k-2) - 1).
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 24*x^3 + 229*x^4 + 2449*x^5 + 28035*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^7 + (A(x)-1)*(A(x)^4-1)/A(x)^17 + (A(x)-1)*(A(x)^4-1)*(A(x)^7-1)/A(x)^30 + (A(x)-1)*(A(x)^4-1)*(A(x)^7-1)*(A(x)^10-1)/A(x)^46 +
(A(x)-1)*(A(x)^4-1)*(A(x)^7-1)*(A(x)^10-1)*(A(x)^13-1)/A(x)^65 +...
Related expansions.
A(x)^2 = 1 + 2*x + 7*x^2 + 54*x^3 + 515*x^4 + 5500*x^5 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 136*x^3 + 1295*x^4 + 13816*x^5 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 190*x^3 + 1810*x^4 + 19316*x^5 +...
where 1+x = A(x)^2 + A(x)^4 - A(x)^5.
PROG
(PARI) /* From 1+x = A(x)^2 + A(x)^4 - A(x)^5: */
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(Ser(A)^2+Ser(A)^4-Ser(A)^5)[#A]); A[n+1]}
for(n=0, 25, print1(a(n) , ", "))
(PARI) /* From Series Reversion: */
{a(n)=local(A=1+serreverse(x - 3*x^2 - 6*x^3 - 4*x^4 - x^5 + x^2*O(x^n))); polcoeff(A, n)}
for(n=0, 25, print1(a(n) , ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 27 2014
STATUS
approved