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A051524 Second unsigned column of triangle A051338. 17


%S 0,1,13,146,1650,19524,245004,3272688,46536624,703404576,11277554400,

%T 191338156800,3427105248000,64651956364800,1281740285145600,

%U 26648514872985600,579892995734169600,13183403757582643200

%N Second unsigned column of triangle A051338.

%C 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 Mitrinovic, D. S. and Mitrinovic, R. S.: see reference given for triangle A051338.

%H G. C. Greubel, <a href="/A051524/b051524.txt">Table of n, a(n) for n = 0..440</a>

%F a(n) = A051338(n, 1)*(-1)^(n-1);

%F E.g.f.: -log(1-x)/(1-x)^6.

%F For n>=1, a(n) = n!*sum((-1)^k*binomial(-6,k)/(n-k),k=0..n-1). - _Milan Janjic_, Dec 14 2008

%F 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 Conjecture: a(n) +(-2*n-9)*a(n-1) +(n+4)^2*a(n-2)=0. - _R. J. Mathar_, Aug 04 2013

%t f[k_] := k + 5; 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. A001725 (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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Last modified January 21 10:26 EST 2020. Contains 331105 sequences. (Running on oeis4.)