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A051524 Second unsigned column of triangle A051338.

%I A051524 #42 Jan 25 2023 13:29:27
%S A051524 0,1,13,146,1650,19524,245004,3272688,46536624,703404576,11277554400,
%T A051524 191338156800,3427105248000,64651956364800,1281740285145600,
%U A051524 26648514872985600,579892995734169600,13183403757582643200
%N A051524 Second unsigned column of triangle A051338.
%C A051524 The asymptotic expansion of the higher order exponential integral E(x,m=2,n=6) ~ exp(-x)/x^2*(1 - 13/x + 146/x^2 - 1650/x^3 + 19524/x^4 - 245004/x^5 + 3272688/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - _Johannes W. Meijer_, Oct 20 2009
%D A051524 Mitrinovic, D. S. and Mitrinovic, R. S.: see reference given for triangle A051338.
%H A051524 G. C. Greubel, <a href="/A051524/b051524.txt">Table of n, a(n) for n = 0..440</a>
%F A051524 a(n) = A051338(n, 1)*(-1)^(n-1);
%F A051524 E.g.f.: -log(1-x)/(1-x)^6.
%F A051524 For n>=1, a(n) = n!*Sum_{k=0..n-1} (-1)^k*binomial(-6,k)/(n-k). - _Milan Janjic_, Dec 14 2008
%F A051524 a(n) = n!*[5]h(n), where [k]h(n) denotes the k-th successive summation of h(n) from 0 to n. - _Gary Detlefs_, Jan 04 2011
%F A051524 Conjecture: a(n) +(-2*n-9)*a(n-1) +(n+4)^2*a(n-2)=0. - _R. J. Mathar_, Aug 04 2013
%t A051524 f[k_] := k + 5; t[n_] := Table[f[k], {k, 1, n}]
%t A051524 a[n_] := SymmetricPolynomial[n - 1, t[n]]
%t A051524 Table[a[n], {n, 1, 16}]
%t A051524 (* _Clark Kimberling_, Dec 29 2011 *)
%Y A051524 Cf. A001725 (first unsigned column).
%Y A051524 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 A051524 easy,nonn
%O A051524 0,3
%A A051524 _Wolfdieter Lang_