This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A121353 #39 Sep 15 2024 02:04:26
%S A121353 0,1,4,27,266,3431,54630,1034539,22705228,566596161,15841987280,
%T A121353 490535009519,16662348336366,616016353436023,24623991789104554,
%U A121353 1058215630578059799,48653295014801646200,2382953240094702604001,123864915189909733761852,6810187382204940654297859
%N A121353 a(n) = (3*n - 2)*a(n-1) - a(n-2) starting a(0)=0, a(1)=1.
%C A121353 In the hypergeometric family a(n) = (a0*n+c0)*a(n-1)+b0*a(n-2) we have A053984, A058797, A121323, A121351, and this here with a0=3, where a(n) can be expressed in a characteristic cross-product of Bessel functions.
%H A121353 Harvey P. Dale, <a href="/A121353/b121353.txt">Table of n, a(n) for n = 0..381</a>
%H A121353 S. Janson, <a href="https://doi.org/10.46298/dmtcs.520">A divergent generating function that can be summed and analysed analytically</a>, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.
%F A121353 a(n) = (Pi/3) * (BesselJ(1/3+n,2/3) * BesselY(1/3,2/3) - BesselJ(1/3,2/3) * BesselY(1/3+n,2/3)).
%F A121353 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-2/3,k+1/3), cf. A058798. - _Peter Bala_, Aug 01 2013
%F A121353 a(n) ~ n! * BesselJ(1/3, 2/3) * 3^(n-2/3) * n^(-2/3). - _Vaclav Kotesovec_, Jul 31 2014
%F A121353 a(n) = 3^n*Gamma(n+1/3)*hypergeometric([1/2-n/2, 1-n/2], [4/3, 2/3-n, 1-n], -4/9)/Gamma(1/3) for n >= 2. - _Peter Luschny_, Sep 10 2014
%t A121353 f[n_Integer] = Module[{a}, a[n] /. RSolve[{a[n] == (3*n - 2)*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 A121353 RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(3n-2)*a[n-1]-a[n-2]}, a, {n, 20}] (* _Vaclav Kotesovec_, Jul 31 2014 *)
%t A121353 nxt[{n_,a_,b_}]:={n+1,b,b(3n+1)-a}; NestList[nxt,{1,0,1},20][[;;,2]] (* _Harvey P. Dale_, Jun 03 2023 *)
%o A121353 (Sage)
%o A121353 def A121353(n):
%o A121353 if n < 2: return n
%o A121353 return 3^n*gamma(n+1/3)*hypergeometric([1/2-n/2,1-n/2], [4/3, 2/3 -n, 1-n], -4/9)/gamma(1/3)
%o A121353 [round(A121353(n).n(100)) for n in (0..19)] # _Peter Luschny_, Sep 10 2014
%Y A121353 Cf. A053984, A058798, A121351, A121323.
%K A121353 nonn,easy
%O A121353 0,3
%A A121353 _Roger L. Bagula_ and Bob Hanlon (hanlonr(AT)cox.net), Sep 05 2006