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E.g.f: exp(x/(1-3*x))/sqrt(1-9*x^2).
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%I #9 Jan 30 2020 21:29:15

%S 1,1,16,100,2116,27556,732736,14776336,476112400,13013333776,

%T 494512742656,17019717246016,747017670477376,30923039616270400,

%U 1542024562112889856,74433082892402872576,4161241771884669788416

%N E.g.f: exp(x/(1-3*x))/sqrt(1-9*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 A115327 which has e.g.f.: exp(x+3/2*x^2).

%F D-finite with recurrence: a(n) = (3*n-2)*a(n-1) - 27*(n-1)*(n-2)^2*a(n-3) + 3*(n-1)*(3*n-2)*a(n-2). - _Vaclav Kotesovec_, Jun 26 2013

%F a(n) ~ 1/2*exp(-1/6+2*sqrt(n/3)-n)*3^n*n^n. - _Vaclav Kotesovec_, Jun 26 2013

%t CoefficientList[Series[E^(x/(1-3*x))/Sqrt[1-9*x^2], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 26 2013 *)

%o (PARI) a(n)=local(m=3);n!*polcoeff(exp(x/(1-m*x+x*O(x^n)))/sqrt(1-m^2*x^2+x*O(x^n)),n)

%Y Cf. A115327.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 20 2006