Result: The Fennessey-Larcombe-French sequence \(\{1,8,144,2432,40000,\dots\}\): formulation and asymptotic form.
Further Information
Fennessey-Larcombe-French sequence \(\{V_n\}_{s=0}^\infty=\{1,8,144,2432,40000,\dots\}\) under consideration arose in connection with the expansion of the transformed integrand of the complete elliptic integral (of the second kind) \(E(c)=\int_0^{\pi/2}\sqrt{1-c^2\sin^2\phi d\phi}=b(\pi/2)\sum_{s=0}^\infty V_s\left[(1-b)/16\right]^s\) employing the substitution \(\theta(\phi)=\tan^{-1}[\sqrt{b}\tan\phi]\) with \(b^2(c)=1-c^2\). The authors prove several formulae for these these numbers (which are integral), establish their asymptotics and show their relation to their counterpart Catalan-Larcombe-French numbers derived in a similar manner starting with the complete elliptic integral \(\int_0^{\pi/2} (1-c^2\sin^2\phi)^{-1/2}\,d \phi\) [ibid. 143, 33--64 (2000; Zbl 0971.05001); 148, 65--91 (2001; Zbl 0999.05003); Util. Math. 60, 67--77 (2001; Zbl 1011.05004)].