A235136 - OEIS

%I #23 Aug 08 2018 07:54:59

%S 1,1,3,5,21,45,231,585,3465,9945,65835,208845,1514205,5221125,

%T 40883535,151412625,1267389585,4996616625,44358635475,184874815125,

%U 1729986783525,7579867420125,74389431691575,341094033905625,3496303289504025,16713607661375625

%N a(n) = (2*n - 1) * a(n-2) for n>1, a(0) = a(1) = 1.

%H G. C. Greubel, <a href="/A235136/b235136.txt">Table of n, a(n) for n = 0..730</a>

%F Let b(n) = a(2*n - 2) / a(2*n + 1). Then b(-n) = b(n), 0 = b(n+1) * (b(n+1) + 2*b(n+2)) + b(n) * (2*b(n+1) - 5*b(n+2)) for all n in Z.

%F a(n-1) + a(n-2) = A196265(n) if n>1.

%F a(2*n) = A008545(n). a(2*n - 1) = A007696(n). a(n) = A007662(2*n - 1).

%F E.g.f. A(x) =: y satisfies 0 = y * 3 + y' * 2*x - y''.

%F 0 = a(n)*(2*a(n+1) - a(n+3)) + a(n+1)*(a(n+2)) for all n in Z. - _Michael Somos_, Jan 24 2014

%F Let b(n) = a(n - 2) / a(n + 1). Then b(-n) = (-1)^n * b(n), 0 = b(n) * (b(n+1) - 4*b(n+3)) + b(n+2) * (2*b(n+1) + b(n+3)) for all n in Z. - _Michael Somos_, Sep 13 2014

%e G.f. = 1 + x + 3*x^2 + 5*x^3 + 21*x^4 + 45*x^5 + 231*x^6 + 585*x^7 + ...

%t a[ n_] := 2^n If[ OddQ[n], 2 Pochhammer[ 1/4, (n + 1)/2], Pochhammer[ 3/4, n/2]]; (* _Michael Somos_, Jan 16 2014 *)

%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (-2 Gamma[5/2] HermiteH[ -3/2, x] + (3 Gamma[5/4] + 2 Gamma[7/4]) Hypergeometric1F1[ 3/4, 1/2, x^2]) / (3 Gamma[5/4]), {x, 0, n}] // FullSimplify]; (* _Michael Somos_, Jan 16 2014 *)

%t RecurrenceTable[{a[0]==a[1]==1, a[n]==(2 n - 1) a[n - 2]}, a, {n, 25}] (* _Vincenzo Librandi_, Aug 08 2018 *)

%o (PARI) {a(n) = if( n<0, (-1)^(-n\2) / a(-1-n), if( n<2, 1, (2*n - 1) * a(n-2)))};

%Y Cf. A007662, A007696, A008545, A196265.

%K nonn

%O 0,3

%A _Michael Somos_, Jan 03 2014