A235136 a(n) = (2*n - 1) * a(n-2) for n>1, a(0) = a(1) = 1.

1, 1, 3, 5, 21, 45, 231, 585, 3465, 9945, 65835, 208845, 1514205, 5221125, 40883535, 151412625, 1267389585, 4996616625, 44358635475, 184874815125, 1729986783525, 7579867420125, 74389431691575, 341094033905625, 3496303289504025, 16713607661375625

Offset: 0

Author

Michael Somos, Jan 03 2014

Keywords

• nonn
• changed

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..730

Crossrefs

Cf. A007662, A007696, A008545, A196265.

Cf. A068465, A068466.

Programs

Mathematica

a[ n_] := 2^n If[ OddQ[n], 2 Pochhammer[ 1/4, (n + 1)/2], Pochhammer[ 3/4, n/2]]; (* Michael Somos, Jan 16 2014 *)
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 *)
RecurrenceTable[{a[0]==a[1]==1, a[n]==(2 n - 1) a[n - 2]}, a, {n, 25}] (* Vincenzo Librandi, Aug 08 2018 *)

PARI

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

Formula

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.

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

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

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

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

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

a(n) ~ c * sqrt(Pi) * (2*n)^(n/2+1/4) / exp(n/2), where c = 2/Gamma(1/4) if n is odd, and 1/Gamma(3/4) if n is even. - Amiram Eldar, Sep 01 2025