A121323 a(n) = (2*n+1)*a(n-1) - a(n-2) starting a(0)=0, a(1)=1.
0, 1, 5, 34, 301, 3277, 42300, 631223, 10688491, 202450106, 4240763735, 97335115799, 2429137131240, 65489367427681, 1896762518271509, 58734148698989098, 1936330144548368725, 67712820910493916277, 2503438043543726533524, 97566370877294840891159
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..300
- S. Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.
Programs
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Maple
A121323 := proc(n) BesselJ(3/2+n,1)*BesselY(3/2,1)-BesselJ(3/2,1)*BesselY(3/2+n,1) ; simplify(Pi*%/2 ); end proc: # R. J. Mathar, Oct 13 2012
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Mathematica
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] 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 *) 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 *) -
Sage
def A121323(n): if n < 2: return n 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)) [round(A121323(n).n(100)) for n in (0..19)] # Peter Luschny, Sep 10 2014
Formula
2*a(n)= Pi*BesselJ_{3/2 + n}(1) * BesselY_{3/2}(1) - Pi*BesselJ_{3/2}(1) *BesselY_{3/2 + n}(1).
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
a(n) ~ (sin(1)-cos(1))*n^(n+1)*2^(n+3/2)/exp(n). - Vaclav Kotesovec, Oct 21 2012
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
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