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 A303058 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1+x)^(n^2) * x^n / A(x)^n. 11
 1, 1, 1, 2, 5, 16, 61, 259, 1228, 6284, 34564, 201978, 1246652, 8084728, 54862377, 388266809, 2857708840, 21822753453, 172550972216, 1410144139982, 11892084248959, 103343300813517, 924223611649636, 8496346816801059, 80201063980292729, 776585923239589681, 7706568335863727817, 78311132374535936605 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..430 FORMULA G.f.: A(x) = 1/(1 - q*x/(A(x) - q*(q^2-1)*x/(1 - q^5*x/(A(x) - q^3*(q^4-1)*x/(1 - q^9*x/(A(x) - q^5*(q^6-1)*x/(1 - q^13*x/(A(x) - q^7*(q^8-1)*x/(1 - ...))))))))), where q = (1+x), a continued fraction due to a partial elliptic theta function identity. G.f.: A(x) = Sum_{n>=0} x^n/A(x)^n * (1+x)^n * Product_{k=1..n} (A(x) - x*(1+x)^(4*k-3)) / (A(x) - x*(1+x)^(4*k-1)), due to a q-series identity. EXAMPLE G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 61*x^6 + 259*x^7 + 1228*x^8 + 6284*x^9 + 34564*x^10 + 201978*x^11 + 1246652*x^12 + ... such that A(x) = 1 + (1+x)*x/A(x) + (1+x)^4*x^2/A(x)^2 + (1+x)^9*x^3/A(x)^3 + (1+x)^16*x^4/A(x)^4 + (1+x)^25*x^5/A(x)^5 + (1+x)^36*x^6/A(x)^6 + ... PROG (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(n=0, #A, ((1+x)^n +x*O(x^#A))^n * x^n/Ser(A)^n ) )[#A] ); A[n+1]} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A303057, A303290, A301929, A321607, A321608. Sequence in context: A307228 A104858 A351143 * A322616 A178123 A138265 Adjacent sequences: A303055 A303056 A303057 * A303059 A303060 A303061 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 20 2018 STATUS approved

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Last modified June 13 03:07 EDT 2024. Contains 373366 sequences. (Running on oeis4.)