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A051562 Second unsigned column of triangle A051380.

%I A051562 #32 Jan 25 2023 13:29:40
%S A051562 0,1,19,299,4578,71394,1153956,19471500,343976400,6366517200,
%T A051562 123418922400,2503748556000,53091962697600,1175271048201600,
%U A051562 27123099523027200,651708291206649600,16282170039031142400
%N A051562 Second unsigned column of triangle A051380.
%C A051562 The asymptotic expansion of the higher order exponential integral E(x,m=2,n=9) ~ exp(-x)/x^2*(1 - 19/x + 299/x^2 - 4578/x^3 + 71394/x^4 - 1153956/x^5 + 19471500/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - _Johannes W. Meijer_, Oct 20 2009
%D A051562 Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051380.
%H A051562 G. C. Greubel, <a href="/A051562/b051562.txt">Table of n, a(n) for n = 0..440</a>
%F A051562 a(n) = A051380(n, 2)*(-1)^(n-1).
%F A051562 E.g.f.: -log(1-x)/(1-x)^9.
%F A051562 a(n) = n!*Sum_{k=0..n-1} ((-1)^k*binomial(-9,k)/(n-k)), for n>=1. - _Milan Janjic_, Dec 14 2008
%F A051562 a(n) = n!*[8]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
%t A051562 f[k_] := k + 8; t[n_] := Table[f[k], {k, 1, n}]
%t A051562 a[n_] := SymmetricPolynomial[n - 1, t[n]]
%t A051562 Table[a[n], {n, 1, 16}]
%t A051562 (* _Clark Kimberling_, Dec 29 2011 *)
%Y A051562 Cf. A049389 (first unsigned column).
%Y A051562 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 A051562 easy,nonn
%O A051562 0,3
%A A051562 _Wolfdieter Lang_