A107590 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n*(n-1)/2).

1, 1, 1, 2, 5, 15, 50, 181, 698, 2837, 12062, 53374, 244923, 1162536, 5697119, 28786266, 149814059, 802436166, 4420515689, 25031466730, 145616087486, 869760092469, 5330945435272, 33508699787635, 215863606818041

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

G.f. A(x) = x/series-reversion(x*F(x)) and thus A(x) = F(x/A(x)) where F(x) = A(x*F(x)) is the g.f. of A107591.

G.f. A(x)^2 = x/series-reversion(x*G(x)^2) and thus A(x) = G(x/A(x)^2) where G(x) = A(x*G(x)^2) is the g.f. of A107592.

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

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

A = 1/(1- x/(1- (A-1)*x/(1- A^2*x/(1- A*(A^2-1)*x/(1- A^4*x/(1- A^2*(A^3-1)*x/(1- A^6*x/(1- A^3*(A^4-1)*x/(1- ...)))))))))

due to an identity of a partial elliptic theta function.

(End)

Example

A = 1 + x + x^2*A^1 + x^3*A^3 + x^4*A^6 + x^5*A^10 +...
= 1 + x + (x^2 + x^3 + x^4 + 2*x^5 + 5*x^6 + 15*x^7 +...)
+ (x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 33*x^7 +...)
+ (x^4 + 6*x^5 + 21*x^6 + 62*x^7 +...)
+ (x^5 + 10*x^6 + 55*x^7 +...) +...
= 1 + x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 50*x^6 + 181*x^7 +...

Prog

(PARI) {a(n) = my(A=1+x+x*O(x^n)); for(k=1, n, A = 1 + sum(j=1, n, x^j * A^(j*(j-1)/2) + x*O(x^n)) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A107591, A107592. A155804, A219358.

Sequence in context: A337526 A346661 A060049 * A245311 A148367 A306836

Adjacent sequences: A107587 A107588 A107589 * A107591 A107592 A107593

Keyword

eigen,nonn

Author

Paul D. Hanna, May 17 2005

Status

approved