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 A192259 G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 + A(x))^n * A(x)^(n*(n+1)/2). 1
 1, 2, 10, 74, 658, 6514, 69210, 773306, 8974114, 107288162, 1314003882, 16420439978, 208754062258, 2693915486418, 35228738082298, 466239274517274, 6238546207411778, 84330947396776642, 1150982783030893386, 15854319075541606666, 220344302315492953298, 3089322686040279975474, 43693043476823499717018, 63085549664634453982706, 6423320378114329801258421518738 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA Let A = g.f. A(x), then A satisfies: (1) A = Sum_{n>=0} x^n*(1+A)^n*A^n * Product_{k=1..n} (1 - x*(1+A)*A^(2*k-1))/(1 - x*(1+A)*A^(2*k)) (2) A = 1/(1- A*(1+A)*x/(1- A*(A-1)*(1+A)*x/(1- A^3*(1+A)*x/(1- A^2*(A^2-1)*(1+A)*x/(1- A^5*(1+A)*x/(1- A^3*(A^3-1)*(1+A)*x/(1- A^7*(1+A)*x/(1- A^4*(A^4-1)*(1+A)*x/(1- ...))))))))) (continued fraction) The above formulas are due to (1) a q-series identity and (2) a partial elliptic theta function expression. EXAMPLE G.f.: A(x) = 1 + 2*x + 10*x^2 + 74*x^3 + 658*x^4 + 6514*x^5 +... Let A = g.f. A(x), then A satisfies: A = 1 + x*(1+A)*A + x^2*(1+A)^2*A^3 + x^3*(1+A)^3*A^6 + x^4*(1+A)^4*A^10 +... Equivalently, A = 1 + x*(A + A^2) + x^2*(A^3 + 2*A^4 + A^5) + x^3*(A^6 + 3*A^7 + 3*A^8 + A^9) +... PROG (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*(1+A)^m*(A+x*O(x^n))^(m*(m+1)/2))); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A107591, A192260. Sequence in context: A001395 A047853 A151387 * A245901 A141149 A306045 Adjacent sequences:  A192256 A192257 A192258 * A192260 A192261 A192262 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 26 2011 STATUS approved

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Last modified March 21 04:59 EDT 2019. Contains 321364 sequences. (Running on oeis4.)