OFFSET
0,3
COMMENTS
Compare the g.f. to the identity:
G(x) = Sum_{n>=0} 1/G(x)^(2*n) * Product_{k=1..n} (1 - 1/G(x)^(2*k-1))
which holds for all power series G(x) such that G(0)=1.
FORMULA
G.f. satisfies: 1+x = A(y) where y = x - 6*x^2 + 3*x^3 + 61*x^4 + 15*x^5 - 567*x^6 - 1946*x^7 - 3607*x^8 - 4489*x^9 - 4015*x^10 - 2640*x^11 - 1274*x^12 - 441*x^13 - 104*x^14 - 15*x^15 - x^16, which is the g.f. of row 4 in triangle A214690.
G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+8)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).
EXAMPLE
G.f.: A(x) = 1 + x + 6*x^2 + 69*x^3 + 929*x^4 + 13692*x^5 + 213402*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^9 + (A(x)-1)*(A(x)^3-1)/A(x)^20 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^33 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^48 +
(A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^65 +...
PROG
(PARI) {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 6*x^2 + 3*x^3 + 61*x^4 + 15*x^5 - 567*x^6 - 1946*x^7 - 3607*x^8 - 4489*x^9 - 4015*x^10 - 2640*x^11 -
1274*x^12 - 441*x^13 - 104*x^14 - 15*x^15 - x^16 +x^2*O(x^n)), n))}
(PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(8*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 26 2012
STATUS
approved