OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..250
M. Bruschi, F. Calogero and R. Droghei, Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials, J. Physics A, 40(2007), pp. 3815-3829.
FORMULA
a(n) = Sum_{m=0..n} |A129467(n,m)| for n >= 0.
a(n) = Sum_{j=0..n-1} |A130559(n-1, j)|, n >= 1.
For n > 0, a(n) = n! * Product_{k=1..n} [Gamma(k + 1/k)/Gamma(k - 1 + 1/k)]. - Gerald McGarvey, Nov 05 2007
a(n) = Product_{k=0..n} (k^2 - k + 1). - Gary Detlefs, Mar 04 2012
a(n) ~ c*n!*(n-1)! for c = Product_{k>=1} (1+1/(k^2+k)) = 2.428189792... [Charles R Greathouse IV, Mar 04 2012], c = cosh(sqrt(3)*Pi/2)/Pi. - Vaclav Kotesovec, Aug 24 2016
G.f.: 1 + x + 3*x^2/(Q(0)-3*x), where Q(k) = 1 + x*(k^2+3*k+3) - x*(k^2+5*k+7)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 15 2013
MATHEMATICA
Round@Table[Cosh[Sqrt[3] Pi/2] Gamma[n + 1/2 + I Sqrt[3]/2] Gamma[n + 1/2 - I Sqrt[3]/2]/Pi, {n, 0, 20}] (* Vladimir Reshetnikov, Aug 23 2016 *)
Product[k^2-k+1, {k, 0, Range[0, 30]}] (* G. C. Greubel, Feb 10 2024 *)
PROG
(PARI) a(n)=prod(k=1, n, k^2-k+1) \\ Charles R Greathouse IV, Mar 04 2012
(Magma) [1] cat [n le 1 select 1 else (n^2-n+1)*Self(n-1): n in [1..30]]; // G. C. Greubel, Feb 10 2024
(SageMath)
[A130032(n) for n in range(31)] # G. C. Greubel, Feb 10 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, May 04 2007
EXTENSIONS
Definition corrected by Wolfdieter Lang, Jun 04 2010
STATUS
approved