A051560 - OEIS

%I #38 Jan 25 2023 13:29:35

%S 0,1,17,242,3382,48504,725592,11393808,188204400,3270729600,

%T 59753750400,1146140409600,23046980025600,485075533132800,

%U 10669304848204800,244861798361241600,5854837379724748800

%N Second unsigned column of triangle A051379.

%C The asymptotic expansion of the higher order exponential integral E(x,m=2,n=8) ~ exp(-x)/x^2*(1 - 17/x + 242/x^2 - 3382/x^3 + 48504/x^4 - 725592/x^5 + 11393808/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - _Johannes W. Meijer_, Oct 20 2009

%D Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051379.

%H G. C. Greubel, <a href="/A051560/b051560.txt">Table of n, a(n) for n = 0..440</a>

%F a(n) = A051379(n, 2)*(-1)^(n-1).

%F E.g.f.: -log(1-x)/(1-x)^8.

%F a(n) = n!*Sum_{k=0..n-1} ((-1)^k*binomial(-8,k)/(n-k)), for n>=1. - _Milan Janjic_, Dec 14 2008

%F a(n) = n!*[7]h(n), where [k]h(n) denotes the k-th successive summation of the harmonic numbers from 0 to n. - _Gary Detlefs_, Jan 04 2011

%F Conjecture: a(n) +(-2*n-13)*a(n-1) +(n+6)^2*a(n-2)=0. - _R. J. Mathar_, Aug 04 2013

%t f[k_] := k + 7; t[n_] := Table[f[k], {k, 1, n}]

%t a[n_] := SymmetricPolynomial[n - 1, t[n]]

%t Table[a[n], {n, 1, 16}]

%t (* _Clark Kimberling_, Dec 29 2011 *)

%Y Cf. A049388 (first unsigned column).

%Y Related to n!*the k-th successive summation of the harmonic numbers: k=0..A000254, k=1..A001705, k= 2..A001711, k=3..A001716, k=4..A001721, k=5..A051524, k=6..A051545, k=7..A051560, k=8..A051562, k=9..A051564. - _Gary Detlefs_ Jan 04 2011

%K easy,nonn

%O 0,3

%A _Wolfdieter Lang_