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E.g.f: exp(x/(1-4*x))/sqrt(1-16*x^2).
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%I #9 Jan 30 2020 21:29:15

%S 1,1,25,169,5329,78961,3031081,71995225,3275616289,108078535009,

%T 5707994717881,241468632426121,14559189135946225,750901957984356049,

%U 50993055118129961929,3098872535897951163961,234376094007794247481921

%N E.g.f: exp(x/(1-4*x))/sqrt(1-16*x^2).

%C Term-by-term square of sequence with e.g.f.: exp(x+m/2*x^2) is given by e.g.f.: exp(x/(1-m*x))/sqrt(1-m^2*x^2) for all m.

%F Equals term-by-term square of A115329 which has e.g.f.: exp(x+2*x^2).

%F D-finite with recurrence: a(n) = (4*n-3)*a(n-1) + 4*(n-1)*(4*n-3)*a(n-2) - 64*(n-1)*(n-2)^2*a(n-3). - _Vaclav Kotesovec_, Jun 26 2013

%F a(n) ~ 2^(2*n-1)*n^n*exp(sqrt(n)-n-1/8) * (1 + 25/(96*sqrt(n))). - _Vaclav Kotesovec_, Jun 26 2013

%t CoefficientList[Series[E^(x/(1-4*x))/Sqrt[1-16*x^2], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 26 2013 *)

%o (PARI) a(n)=local(m=4);n!*polcoeff(exp(x/(1-m*x+x*O(x^n)))/sqrt(1-m^2*x^2+x*O(x^n)),n)

%Y Cf. A115329.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 20 2006