OFFSET
0,2
COMMENTS
A special case of an integral in Comtet (1967, pp. 85-86) yields
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.
LINKS
Louis Comtet, Fonctions génératrices et calcul de certaines intégrales, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87; see pp. 81-83.
Petros Hadjicostas, Proof of the claim a(n) = n!*A063079(n+1)/A060818(n), 2020.
V. H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
FORMULA
a(n) = binomial(4*n+2, 2*n+1)*n!/2^(n+1).
a(n) = n!*Sum_{j=0..n} 2^(n-2*j)*binomial(2*n+1,2*j)*binomial(2*j,j).
E.g.f.: 2/(sqrt(1 - 8*s) * (sqrt(1 + sqrt(8*s)) + sqrt(1 - sqrt(8*s)))).
E.g.f.: sqrt(2/(1 + sqrt(1 - 8*s))/(1 - 8*s)).
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Petros Hadjicostas, May 15 2020
STATUS
approved