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%I #36 Jan 25 2023 13:29:45
%S 0,1,21,362,6026,101524,1763100,31813200,598482000,11752855200,
%T 240947474400,5154170774400,114942011990400,2669517204076800,
%U 64496340380102400,1619153396908185600,42188624389562112000
%N Second unsigned column of triangle A051523.
%C The asymptotic expansion of the higher order exponential integral E(x,m=2,n=10) ~ exp(-x)/x^2*(1 - 21/x + 362/x^2 - 6026/x^3 + 101524/x^4 - 1763100/x^5 + 31813200/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 A051523.
%H G. C. Greubel, <a href="/A051564/b051564.txt">Table of n, a(n) for n = 0..440</a>
%F a(n) = A051523(n, 2)*(-1)^(n-1).
%F E.g.f.: -log(1-x)/(1-x)^10.
%F a(n) = n!*Sum_{k=0..n-1}((-1)^k*binomial(-10,k)/(n-k)), for n>=1. - _Milan Janjic_, Dec 14 2008
%F a(n) = n!*[9]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 f[n_] := n!*Sum[(-1)^k*Binomial[-10, k]/(n - k), {k, 0, n - 1}]; Array[f, 17, 0]
%t Range[0, 16]! CoefficientList[ Series[-Log[(1 - x)]/(1 - x)^10, {x, 0, 16}], x]
%t (* Or, using elementary symmetric functions: *)
%t f[k_] := k + 9; 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. A049398 (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_
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