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A106174
a(n) = 2*n*a(n-1) - a(n-2), with a(0)=0, a(1)=1.
5
0, 1, 4, 23, 180, 1777, 21144, 294239, 4686680, 84066001, 1676633340, 36801867479, 881568186156, 22883970972577, 639869619046000, 19173204600407423, 612902677593991536, 20819517833595304801, 748889739331836981300
OFFSET
0,3
COMMENTS
Bessel recurrence at x=1: J(x,n) = (2*n/x)*J(x,n-1) - J(x,n-2).
REFERENCES
Abramowitz and Stegun, Handbook of Mathematical Functions, 9th printing, 1972, page 385.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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.
FORMULA
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,k+1), cf. A058798. - Peter Bala, Aug 01 2013
a(n) = n!*2^(n-1)*hypergeometric2F3([(1-n)/2, 1-n/2],[2, 1-n, -n], -1) for n >= 2. - Peter Luschny, Sep 10 2014
From Vaclav Kotesovec, Jun 10 2019: (Start)
a(n) = Pi*(BesselJ(1 + n, 1)*BesselY(1, 1) - BesselJ(1, 1)*BesselY(1 + n, 1))/2.
a(n) ~ BesselJ(1,1) * 2^n * n!. (End)
MATHEMATICA
F[0]=0; F[1]=1; F[n_]:= F[n]= 2*n*F[n-1]-F[n-2]; Table[F[n], {n, 0, 20}]
RecurrenceTable[{a[0]==0, a[1]==1, a[n]==2n a[n-1]-a[n-2]}, a, {n, 20}] (* Harvey P. Dale, Oct 17 2016 *)
PROG
(Sage)
def A058798(n):
if n < 2: return n
return factorial(n)*2^(n-1)*hypergeometric([1/2-n/2, 1-n/2], [2, 1-n, -n], -1)
[round(A058798(n).n(164)) for n in (0..18)] # Peter Luschny, Sep 10 2014
(PARI) m=20; v=concat([0, 1], vector(m-2)); for(n=3, m, v[n]=2*(n-1)*v[n-1]-v[n-2]); v \\ G. C. Greubel, Mar 25 2019
(Magma) I:=[0, 1]; [n le 2 select I[n] else 2*(n-1)*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Mar 25 2019
(GAP) a:=[0, 1];; for n in [3..20] do a[n]:=2*(n-1)*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Mar 25 2019
CROSSREFS
Cf. A058798.
Sequence in context: A220353 A089465 A220214 * A056814 A058863 A192840
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula, Mar 14 2006
STATUS
approved