The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A303290 G.f. A(x) satisfies: 2 = Sum_{n>=0} (1/2^n) * (1+x)^(n^2) / A(x)^n. 5
 1, 3, 15, 225, 6003, 223029, 10403175, 577700889, 37009173207, 2679339499305, 216031850406327, 19187294118006057, 1861057604220294591, 195742656849628038465, 22192660352433291780159, 2698458809215198981964481, 350326879575505922875480047, 48370384900519379918253881361, 7078145146554395463373624118319, 1094300840117324691452685873392145 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..200 FORMULA G.f.: 2 = 1/(1 - q/(2*A(x) - q*(q^2-1)/(1 - q^5/(2*A(x) - q^3*(q^4-1)/(1 - q^9/(2*A(x) - q^5*(q^6-1)/(1 - q^13/(2*A(x) - q^7*(q^8-1)/(1 - ...))))))))), where q = (1+x), a continued fraction due to a partial elliptic theta function identity. G.f.: 2 = Sum_{n>=0} (1+x)^n/(2^n*A(x)^n) * Product_{k=1..n} (2*A(x) - (1+x)^(4*k-3)) / (2*A(x) - (1+x)^(4*k-1)), due to a q-series identity. a(n) ~ c * 2^(2*n) * n^n / (exp(n) * log(2)^(2*n)), where c = 0.339650521725496... - Vaclav Kotesovec, Oct 06 2020 EXAMPLE G.f.: A(x) = 1 + 3*x + 15*x^2 + 225*x^3 + 6003*x^4 + 223029*x^5 + 10403175*x^6 + 577700889*x^7 + 37009173207*x^8 + 2679339499305*x^9 + 216031850406327*x^10 + ... such that A = A(x) satisfies: 2 = 1 + (1+x)/(2*A) + (1+x)^4/(2*A)^2 + (1+x)^9/(2*A)^3 + (1+x)^16/(2*A)^4 + (1+x)^25/(2*A)^5 + (1+x)^36/(2*A)^6 + (1+x)^49/(2*A)^7 + (1+x)^64/(2*A)^8 + ... PROG (PARI) /* Find A(x) that satisfies the continued fraction: */ {a(n) = my(A=[1], q=1+x, CF=1); for(i=1, n, A=concat(A, 0); m=#A; for(k=0, m, CF = 1/(1 - q^(4*m-4*k+1)/(2*Ser(A) - q^(2*m-2*k+1)*(q^(2*m-2*k+2) - 1)*CF)) ); A[#A] = Vec(CF)[#A]/2 ); A[n+1]} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A303291, A303292, A303058. Sequence in context: A126455 A136466 A296856 * A298114 A301457 A114735 Adjacent sequences:  A303287 A303288 A303289 * A303291 A303292 A303293 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 20 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 8 22:49 EST 2021. Contains 349596 sequences. (Running on oeis4.)