A081405 - OEIS

%I #37 Sep 08 2022 08:45:09

%S 1,1,3,4,15,24,105,192,945,1920,10395,23040,135135,322560,2027025,

%T 5160960,34459425,92897280,654729075,1857945600,13749310575,

%U 40874803200,316234143225,980995276800,7905853580625,25505877196800,213458046676875

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

%C A001147 and A002866 combined.

%H G. C. Greubel, <a href="/A081405/b081405.txt">Table of n, a(n) for n = 0..790</a>

%F a(0)=a(1)=1; a(2n) = A001147(2*n-2) odd terms, double factorial numbers; a(2n-1) = A002866(n) = 2^(n-1)*n!

%F 0 = a(n)*(a(n+1) - a(n+3)) + a(n+1)*a(n+2) if n>=0. - _Michael Somos_, Jan 24 2014

%F a(n) = (n-1)-st term of column 1 of the array at A249159, for n >= 0. - _Clark Kimberling_, Oct 23 2014

%e G.f. = 1 + x + 3*x^2 + 4*x^3 + 15*x^4 + 24*x^5 + 105*x^6 + 192*x^7 + ...

%p a[0]:=1:a[1]:=1:for n from 2 to 50 do a[n]:=(a[n-2]*(n+1)^2) od: seq(sqrt(a[n]), n=0..26); # _Zerinvary Lajos_, Mar 04 2008

%t f[n_]:= (n+1)*f[n-2]; f[0] = 1; f[1] = 1; Table[f[n], {n, 1, 30}]

%t a[ n_]:= If[ n < 0, 0, If[OddQ[n], 2^((n-1)/2) ((n+1)/2)!, (n+1)!!]]; (* _Michael Somos_, Jan 24 2014 *)

%t RecurrenceTable[{a[0]==a[1]==1,a[n]==(n+1)a[n-2]},a,{n,30}] (* _Harvey P. Dale_, Nov 05 2021 *)

%o (PARI) {a(n) = if( n<2, n>=0, (n+1) * a(n-2))}; /* _Michael Somos_, Jan 24 2014 */

%o (PARI) {a(n) = if( n<0, 0, if( n%2, 2^(n\2) * (n\2 + 1)!, (n+1)! / (2^(n\2) * (n\2)!)))}; /* _Michael Somos_, Jan 24 2014 */

%o (Magma) [n le 1 select 1 else (n+1)*Self(n-1): n in [0..30]]; // _Vincenzo Librandi_, Oct 26 2014

%o (Sage)

%o def a(n):

%o     if n<2: return 1

%o     else: return (n+1)*a(n-2)

%o [a(n) for n in (0..30)] # _G. C. Greubel_, Aug 24 2019

%o (GAP)

%o a:= function(n)

%o     if n<2 then return 1;

%o     else return (n+1)*a(n-2);

%o     fi;

%o   end;

%o List([0..30], n-> a(n) ); # _G. C. Greubel_, Aug 24 2019

%Y Cf. A000142, A001147, A002866.

%K nonn

%O 0,3

%A _Labos Elemer_, Apr 01 2003