A214694 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(8*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

1, 1, 6, 69, 929, 13692, 213402, 3456450, 57585400, 980408857, 16982002433, 298322996205, 5302587890821, 95196447689434, 1723782813066284, 31447947375375315, 577509675356805547, 10667460556561578780, 198074286156460874227, 3695152948440645726312

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.

Links

Table of n, a(n) for n=0..19.

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

Cf. A214690, A214692, A214693, A214695, A181998 (variant).

Sequence in context: A198699 A346938 A258792 * A347674 A234509 A177751

Adjacent sequences: A214691 A214692 A214693 * A214695 A214696 A214697

Keyword

nonn

Author

Paul D. Hanna, Jul 26 2012

Status

approved