A130032 Row sums of unsigned triangle A129467.
1, 1, 3, 21, 273, 5733, 177723, 7642089, 435599073, 31798732329, 2893684641939, 321198995255229, 42719466368945457, 6706956219924436749, 1227372988246171925067, 258975700519942276189137, 62413143825306088561582017, 17038788264308562177311890641
Offset: 0
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.
Programs
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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
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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 *) -
PARI
a(n)=prod(k=1,n,k^2-k+1) \\ Charles R Greathouse IV, Mar 04 2012
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SageMath
def A130032(n): return 1 if n<2 else (n^2-n+1)*A130032(n-1) [A130032(n) for n in range(31)] # G. C. Greubel, Feb 10 2024
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
Extensions
Definition corrected by Wolfdieter Lang, Jun 04 2010