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%I #15 Aug 08 2021 11:15:44
%S 1,1,7,19,159,729,7407,48231,581535,4922325,68891175,718638075,
%T 11465661375,142257791025,2550046679775,36691916525775,
%U 730304613424575,11958031070311725,261722208861516375,4805774015579971875,114729101737416849375,2334996696935363855625
%N a(n) = a(n-1) + n*(n+1)*a(n-2) with a(0)=1, a(1)=1.
%F a(n) ~ n! * (Pi - 2) * n^(3/2) / sqrt(2*Pi).
%F a(n) ~ (Pi - 2) * n^(n+2) / exp(n).
%F E.g.f. A(x) satisfies the differential equation -6*A(x) - (6*x + 1)*A'(x) + (1 - x^2)*A''(x) = 0, A(0)=1, A'(0)=1.
%F E.g.f.: (-2 + Pi + 2*Pi*x + 4*sqrt(1-x^2) + 2*x*(-2+sqrt(1-x^2)) - 4*(1+2*x) * arcsin(sqrt(1-x)/sqrt(2))) / (2*(1-x)^(5/2) * (1+x)^(3/2)).
%t RecurrenceTable[{a[n] == a[n-1] + n*(n+1)*a[n-2], a[0]==1, a[1]==1}, a, {n,0,20}]
%t nmax = 20; CoefficientList[Series[(-2 + Pi + 2*Pi*x + 4*Sqrt[1 - x^2] + 2*x*(-2 + Sqrt[1 - x^2]) - 4*(1 + 2*x) * ArcSin[Sqrt[1 - x]/Sqrt[2]]) / (2*(1 - x)^(5/2) * (1 + x)^(3/2)), {x, 0, nmax}], x] * Range[0, nmax]!
%Y Cf. A000246, A000932, A024167.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Aug 08 2021, following a suggestion from _John M. Campbell_
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