A107596 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n^2+n).

1, 1, 3, 14, 80, 514, 3567, 26153, 199900, 1579107, 12816020, 106421359, 901430144, 7771535382, 68085001080, 605420138920, 5459655601753, 49904765136264, 462228258349278, 4337787743946224, 41249375376404380, 397572319756235577

Offset: 0

Author

Paul D. Hanna, May 17 2005

Keywords

• eigen
• nonn

Examples

			A = 1 + x*A^2 + x^2*A^6 + x^3*A^12 + x^4*A^20 + x^5*A^30 ...
= 1 + (x + 2*x^2 + 7*x^3 + 34*x^4 + 197*x^5 + 1272*x^6 +...)
+ (x^2 + 6*x^3 + 33*x^4 + 194*x^5 + 1230*x^6 +...)
+ (x^3 + 12*x^4 + 102*x^5 + 784*x^6 +...)
+ (x^4 + 20*x^5 + 250*x^6 +...) +...
= 1 + x + 3*x^2 + 14*x^3 + 80*x^4 + 514*x^5 + 3567*x^6 +...
		

Crossrefs

Cf. A107592, A107594, A107595.

Programs

PARI

{a(n)=local(A=1+x+x*O(x^n)); for(k=1,n,A=1+sum(j=1,n,x^j*A^(j^2+j)+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. A(x)^2 = (1/x)*series-reversion(x/G107594(x)^2) and thus A(x) = G107594(x*A(x)^2) where G107594(x) is the g.f. of A107594. G.f. A(x) = (1/x)*series-reversion(x/G107595(x)) and thus A(x) = G107595(x*A(x)) where G107595(x) is the g.f. of A107595.

Contribution from Paul D. Hanna, Apr 25 2010: (Start)

Let A = g.f. A(x), then A satisfies the continued fraction:

A = 1/(1- A^2*x/(1- (A^4-A^2)*x/(1- A^6*x/(1- (A^8-A^4)*x/(1- A^10*x/(1- (A^12-A^6)*x/(1- A^14*x/(1- (A^16-A^8)*x/(1- A^18*x)))))))))

due to an identity of a partial elliptic theta function.

(End)