A130033 - OEIS

%I #21 Feb 10 2024 04:02:13

%S 1,-20,508,-17544,808848,-48405888,3663035136,-342678781440,

%T 38879803008000,-5263815891456000,838682139211776000,

%U -155393459730173952000,33136711787903754240000,-8059211591488628981760000,2217755736675770074398720000

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

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

%H G. C. Greubel, <a href="/A130033/b130033.txt">Table of n, a(n) for n = 0..250</a>

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

%F 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

%F 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

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

%t A130033[n_]:= (-1)^n*((n+2)!)^2*(2*(n+2) -(n+3)*HarmonicNumber[n+2,2]);

%t Table[A130033[n], {n,0,30}] (* _G. C. Greubel_, Feb 10 2024 *)

%o (Magma)

%o h:= func< n,k | (&+[1/j^k : j in [1..n]]) >;

%o A130033:= func< n | (-1)^n*(Factorial(n+2))^2*(2*(n+2) - (n+3)*h(n+2,2)) >;

%o [A130033(n): n in [0..30]]; // _G. C. Greubel_, Feb 10 2024

%o (SageMath)

%o def A130033(n): return (-1)^n*(factorial(n+2))^2*(2*(n+2) - (n+3)*(zeta(2) - psi(1,n+3)))

%o [A130033(n) for n in range(31)] # _G. C. Greubel_, Feb 10 2024

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

%Y Cf. A129467.

%K sign,easy

%O 0,2

%A _Wolfdieter Lang_, May 04 2007