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

0,3

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)
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

Crossrefs

Cf. A130031 (signed row sums), A130559 (unsigned row sums).

Sequence in context: A227820 A336809 A066206 * A174967 A126461 A370741

Adjacent sequences: A130029 A130030 A130031 * A130033 A130034 A130035

Keyword

nonn,easy

Author

Wolfdieter Lang, May 04 2007

Extensions

Definition corrected by Wolfdieter Lang, Jun 04 2010

Status

approved