A257859 a(n) = (2*n-1)*a(n-1) - a(n-2) with a(0)=2, a(1)=1.

2, 1, 1, 4, 27, 239, 2602, 33587, 501203, 8486864, 160749213, 3367246609, 77285922794, 1928780823241, 51999796304713, 1506065312013436, 46636024876111803, 1537482755599676063, 53765260421112550402, 1987777152825564688811, 77469543699775910313227

Offset

0,1

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = 2*A053983(n) - A053984(n).

a(n) = (SBY(0,1) - 2*SBY(1,1))*SBJ(n,1) - (SBJ(0,1) - 2*SBJ(1,1))*SBY(n,1), where SBJ and SBY are the spherical Bessel functions of first and second kind, respectively.

E.g.f.: (2*cos(1-sqrt(1-2*x)) - sin(1-sqrt(1-2*x)))/sqrt(1-2*x).

Sum_{n=0..infinity} a(n-1) x^n/n! = cos(1-sqrt(1-2*x)) + 2*sin(1-sqrt(1-2*x)), where a(-1)=1, a(0)=2, a(1)=1.

a(n) ~ (2*cos(1)-sin(1)) * 2^(n+1/2) * n^n / exp(n). - Vaclav Kotesovec, May 20 2015

Mathematica

RecurrenceTable[{a[0] == 2, a[1] == 1, a[n] == -a[n - 2] + (2 n - 1) a[n - 1]}, a, {n, 30}]

Prog

(Magma) [n le 2 select 3-n else (2*n-3)*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, May 12 2015

Crossrefs

Cf. A053983, A053984.

Sequence in context: A127966 A326570 A184880 * A231622 A165623 A264878

Adjacent sequences: A257856 A257857 A257858 * A257860 A257861 A257862

Keyword

nonn,easy

Author

G. C. Greubel, May 10 2015

Status

approved