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URL: https://oeis.org/A157134

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A157134
G.f. satisfies: A(x) = Sum_{n>=0} x^(n^2) * A(x)^n.
6
1, 1, 1, 1, 2, 4, 7, 11, 18, 33, 63, 117, 211, 383, 713, 1348, 2547, 4793, 9039, 17165, 32785, 62761, 120243, 230768, 444119, 857015, 1656931, 3207990, 6219994, 12079544, 23496417, 45767352, 89256038, 174269488, 340646238, 666604642
OFFSET
0,5
FORMULA
G.f. satisfies: A(x) = B(x/A(x)) where B(x) = A(x*B(x)) = g.f. of A157135,
where A157135(n) = [x^n] A(x)^(n+1)/(n+1) for n>=0,
and a(n) = [x^n] -1/B(x)^(n-1)/(n-1) for n>1.
From Paul D. Hanna, Apr 25 2010: (Start)
G.f. A(x) satisfies the continued fraction:
A(x) = 1/(1- x*A(x)/(1- (x^3-x)*A(x)/(1- x^5*A(x)/(1- (x^7-x^3)*A(x)/(1- x^9*A(x)/(1- (x^11-x^5)*A(x)/(1- x^13*A(x)/(1- (x^15-x^7)*A(x)/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
(End)
From Paul D. Hanna, May 05 2010: (Start)
Let A = g.f. A(x) at x=q, then A satisfies the q-series:
A = Sum_{n>=0} q^n*A^n*Product_{k=1..n} (1-q^(4k-3)*A)/(1-q^(4k-1)*A).
(End)
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 11*x^7 +...
A(x)^2 = 1 + 2*x + 3*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 27*x^6 +...
A(x)^3 = 1 + 3*x + 6*x^2 + 10*x^3 + 18*x^4 + 36*x^5 + 73*x^6 +...
A(x)^4 = 1 + 4*x + 10*x^2 + 20*x^3 + 39*x^4 + 80*x^5 + 168*x^6 +...
where
A(x) = 1 + x*A(x) + x^4*A(x)^2 + x^9*A(x)^3 + x^16*A(x)^4 +...
PROG
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, (A=sum(m=0, sqrtint(n), x^(m^2)*A^m))); polcoeff(A, n)}
CROSSREFS
Cf. A107595. [From Paul D. Hanna, Apr 25 2010]
Sequence in context: A000570 A239552 A023426 * A357932 A127926 A078513
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 24 2009
EXTENSIONS
Typo in data corrected by D. S. McNeil, Aug 17 2010
STATUS
approved