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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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS 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. Contribution 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) Contribution 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. A157135, A157133, A157136. Cf. A107595. [From Paul D. Hanna, Apr 25 2010] Sequence in context: A000570 A239552 A023426 * A127926 A078513 A170804 Adjacent sequences:  A157131 A157132 A157133 * A157135 A157136 A157137 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 24 2009 EXTENSIONS Typo in data corrected by D. S. McNeil, Aug 17 2010 STATUS approved

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