A130033 Fourth (m=3) column sequence of triangle A129467.

1, -20, 508, -17544, 808848, -48405888, 3663035136, -342678781440, 38879803008000, -5263815891456000, 838682139211776000, -155393459730173952000, 33136711787903754240000, -8059211591488628981760000, 2217755736675770074398720000

Offset: 0

Author

Wolfdieter Lang, May 04 2007

Keywords

• sign
• easy

Comments

See the M. Bruschi et al. reference given in A129467.

Examples

			a(3)=-det([20,1,0],[292,40,1],[3824,1092,70])=-17544. [_Mircea Merca_, Apr 06 2013]
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A010790 (m=1 column unsigned), A084915 (m=2 column unsigned).

Cf. A129467.

Programs

Magma

h:= func< n,k | (&+[1/j^k : j in [1..n]]) >;
A130033:= func< n | (-1)^n*(Factorial(n+2))^2*(2*(n+2) - (n+3)*h(n+2,2)) >;
[A130033(n): n in [0..30]]; // G. C. Greubel, Feb 10 2024

Mathematica

A130033[n_]:= (-1)^n*((n+2)!)^2*(2*(n+2) -(n+3)*HarmonicNumber[n+2,2]);
Table[A130033[n], {n,0,30}] (* G. C. Greubel, Feb 10 2024 *)

SageMath

def A130033(n): return (-1)^n*(factorial(n+2))^2*(2*(n+2) - (n+3)*(zeta(2) - psi(1,n+3)))
[A130033(n) for n in range(31)] # G. C. Greubel, Feb 10 2024

Formula

a(n) = A129467(n+3,3),n>=0.

a(n) = (-1)^n*det(PS(i+3,j+2), 1 <= i,j <= n), where PS(n,k) are Legendre-Stirling numbers of the second kind (A071951). - Mircea Merca, Apr 06 2013

a(n) = (-1)^n * ((n+2)!)^2 * (2*(n+2) - (n+3)*h(n+2, 2)), where h(n,k) = Sum_{j=1..n} 1/j^k is the generalized harmonic number. - G. C. Greubel, Feb 10 2024