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A334907 Comtet's expansion of the e.g.f. (sqrt(1 + sqrt(8*s)) - sqrt(1 - sqrt(8*s)))/ sqrt(8*s * (1 - 8*s)).

%I A334907 #49 May 25 2020 05:08:58
%S A334907 1,5,63,1287,36465,1322685,58503375,3053876175,183771489825,
%T A334907 12525477859125,953725671273375,80237355387564375,7391465178302430225,
%U A334907 739967791738943292525,79993069900054731795375,9286937373235386442953375,1152424501315118408602850625
%N A334907 Comtet's expansion of the e.g.f. (sqrt(1 + sqrt(8*s)) - sqrt(1 - sqrt(8*s)))/ sqrt(8*s * (1 - 8*s)).
%C A334907 A special case of an integral in Comtet (1967, pp. 85-86) yields
%C A334907 Integral_{t=-oo..oo} dx/(x^2 + t^2)^(2*n) = Pi * a(n-1)/((n-1)! * 2^(3*n - 2) * t^(4*n-1)) for n >= 1 and t > 0. This integral also follows from a theorem in Moll (2002, p. 312, set a=1), but it requires the summation formula for a(n) shown below.
%H A334907 Louis Comtet, <a href="https://www.jstor.org/stable/43667287">Fonctions génératrices et calcul de certaines intégrales</a>, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87; see pp. 81-83.
%H A334907 Petros Hadjicostas, <a href="/A334907/a334907.pdf">Proof of the claim a(n) = n!*A063079(n+1)/A060818(n)</a>, 2020.
%H A334907 V. H. Moll, <a href="http://www.ams.org/notices/200203/fea-moll.pdf">The evaluation of integrals: a personal story</a>, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
%F A334907 a(n) = binomial(4*n+2, 2*n+1)*n!/2^(n+1).
%F A334907 a(n) = n!*A063079(n+1)/A060818(n) = n!*A001790(2*n+1)/A060818(n) (see the link for a proof).
%F A334907 a(n) = n!*Sum_{j=0..n} 2^(n-2*j)*binomial(2*n+1,2*j)*binomial(2*j,j).
%F A334907 a(n) = 2^n*n!*Sum_{k=0..n} A223549(n,k)/A223550(n,k).
%F A334907 E.g.f.: 2/(sqrt(1 - 8*s) * (sqrt(1 + sqrt(8*s)) + sqrt(1 - sqrt(8*s)))).
%F A334907 E.g.f.: sqrt(2/(1 + sqrt(1 - 8*s))/(1 - 8*s)).
%F A334907 D-finite with recurrence (2*n+1)*a(n) -(4*n-1)*(4*n+1)*a(n-1)=0. - _R. J. Mathar_, May 25 2020
%Y A334907 Cf. A001790, A060818, A063079, A223549, A223550.
%K A334907 nonn
%O A334907 0,2
%A A334907 _Petros Hadjicostas_, May 15 2020