A121323 - OEIS

%I #35 Jan 31 2024 09:49:49

%S 0,1,5,34,301,3277,42300,631223,10688491,202450106,4240763735,

%T 97335115799,2429137131240,65489367427681,1896762518271509,

%U 58734148698989098,1936330144548368725,67712820910493916277,2503438043543726533524,97566370877294840891159

%N a(n) = (2*n+1)*a(n-1) - a(n-2) starting a(0)=0, a(1)=1.

%H Vincenzo Librandi, <a href="/A121323/b121323.txt">Table of n, a(n) for n = 0..300</a>

%H 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 2*a(n)= Pi*BesselJ_{3/2 + n}(1) * BesselY_{3/2}(1) - Pi*BesselJ_{3/2}(1) *BesselY_{3/2 + n}(1).

%F E.g.f.: ((sqrt(1-2*x)+1)*sin(1-sqrt(1-2*x))+(sqrt(1-2*x)-1)*cos(1-sqrt(1-2*x)))/(1-2*x)^(3/2). - _Vaclav Kotesovec_, Oct 21 2012

%F a(n) ~ (sin(1)-cos(1))*n^(n+1)*2^(n+3/2)/exp(n). - _Vaclav Kotesovec_, Oct 21 2012

%F a(n) = Sum_{k = 0..floor((n-1)/2)} (-1)^k*2^(n-2*k-1)*(n-2*k-1)!*binomial(n-k-1,k)*binomial(n-k+1/2,k+3/2), cf. A058798. - _Peter Bala_, Aug 01 2013

%F a(n) = 2^(n+1)*Gamma(n+3/2)*hypergeometric([1/2-n/2, 1-n/2], [5/2, -n-1/2, 1-n], -1)/(3*sqrt(Pi)) for n >= 2. - _Peter Luschny_, Sep 10 2014

%p A121323 := proc(n)

%p     BesselJ(3/2+n,1)*BesselY(3/2,1)-BesselJ(3/2,1)*BesselY(3/2+n,1) ;

%p     simplify(Pi*%/2 );

%p end proc: # _R. J. Mathar_, Oct 13 2012

%t f[n_Integer] = Module[{a}, a[n] /. RSolve[{a[n] == (2*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 CoefficientList[Series[((Sqrt[1-2*x]+1)*Sin[1-Sqrt[1-2*x]]+(Sqrt[1-2*x]-1)*Cos[1-Sqrt[1-2*x]])/(1-2*x)^(3/2),{x,0,20}],x]*Range[0,20]! (* _Vaclav Kotesovec_, Oct 21 2012 *)

%t nxt[{n_,a_,b_}]:={n+1,b,(2n+3)b-a}; NestList[nxt,{1,0,1},20][[All,2]] (* _Harvey P. Dale_, Sep 04 2021 *)

%o (Sage)

%o def A121323(n):

%o     if n < 2: return n

%o     return 2^(n+1)*gamma(n+3/2)*hypergeometric([1/2-n/2, 1-n/2], [5/2, -n-1/2, 1-n],-1) /(3*sqrt(pi))

%o [round(A121323(n).n(100)) for n in (0..19)] # _Peter Luschny_, Sep 10 2014

%Y Cf. A053984, A121351, A121353, A058798.

%K nonn,easy

%O 0,3

%A _Roger L. Bagula_ and Bob Hanlon (hanlonr(AT)cox.net), Sep 05 2006