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%I #24 Jan 09 2024 12:17:49
%S 0,1,7,69,890,14171,268359,5889727,146974816,4109405121,127244583935,
%T 4322206448669,159794394016818,6387453554224051,274500708437617375,
%U 12620645134576175199,618137110885794967376,32130509120926762128353
%N a(n) = (3*n+1)*a(n-1) - a(n-2), starting a(0)=0, a(1)=1.
%F 3*a(n)= Pi*BesselJ_{4/3 + n}(2/3)* BesselY_{4/3}(2/3) - Pi*BesselJ_{4/3}(2/3) * BesselY_{4/3 + n}(2/3).
%F a(n) = sum {k = 0..floor((n-1)/2)} (-1)^k*3^(n-2*k-1)*(n-2*k-1)!*binomial(n-k-1,k)*binomial(n-k+1/3,k+4/3), cf. A058798. - _Peter Bala_, Aug 01 2013
%F a(n) ~ n! * BesselJ(4/3, 2/3) * 3^(n+1/3) * n^(1/3). - _Vaclav Kotesovec_, Jul 31 2014
%t f[n_Integer] = Module[{a}, a[n] /. RSolve[{a[n] == (3*n + 1)*a[n - 1] - a[n - 2], a[0] == 0, a[1] == 1}, a[n], n][[1]] // FullSimplify] Rationalize[N[Table[f[n], {n, 0, 25}], 100], 0]
%t RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(3n+1)*a[n-1]-a[n-2]}, a, {n, 20}] (* _Vaclav Kotesovec_, Jul 31 2014 *)
%t nxt[{n_,a_,b_}]:={n+1,b,(3n+4)b-a}; NestList[nxt,{1,0,1},20][[All,2]] (* _Harvey P. Dale_, Jun 20 2021 *)
%Y Cf. A053984, A058798, A121323, A121353.
%K nonn,easy
%O 0,3
%A _Roger L. Bagula_ and Bob Hanlon (hanlonr(AT)cox.net), Sep 05 2006
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