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A051560 Second unsigned column of triangle A051379.

%I A051560 #38 Jan 25 2023 13:29:35
%S A051560 0,1,17,242,3382,48504,725592,11393808,188204400,3270729600,
%T A051560 59753750400,1146140409600,23046980025600,485075533132800,
%U A051560 10669304848204800,244861798361241600,5854837379724748800
%N A051560 Second unsigned column of triangle A051379.
%C A051560 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 A051560 Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051379.
%H A051560 G. C. Greubel, <a href="/A051560/b051560.txt">Table of n, a(n) for n = 0..440</a>
%F A051560 a(n) = A051379(n, 2)*(-1)^(n-1).
%F A051560 E.g.f.: -log(1-x)/(1-x)^8.
%F A051560 a(n) = n!*Sum_{k=0..n-1} ((-1)^k*binomial(-8,k)/(n-k)), for n>=1. - _Milan Janjic_, Dec 14 2008
%F A051560 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 A051560 Conjecture: a(n) +(-2*n-13)*a(n-1) +(n+6)^2*a(n-2)=0. - _R. J. Mathar_, Aug 04 2013
%t A051560 f[k_] := k + 7; t[n_] := Table[f[k], {k, 1, n}]
%t A051560 a[n_] := SymmetricPolynomial[n - 1, t[n]]
%t A051560 Table[a[n], {n, 1, 16}]
%t A051560 (* _Clark Kimberling_, Dec 29 2011 *)
%Y A051560 Cf. A049388 (first unsigned column).
%Y A051560 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 A051560 easy,nonn
%O A051560 0,3
%A A051560 _Wolfdieter Lang_