A051564 Second unsigned column of triangle A051523.

0, 1, 21, 362, 6026, 101524, 1763100, 31813200, 598482000, 11752855200, 240947474400, 5154170774400, 114942011990400, 2669517204076800, 64496340380102400, 1619153396908185600, 42188624389562112000

Offset

0,3

Comments

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

References

Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051523.

Links

G. C. Greubel, Table of n, a(n) for n = 0..440

Formula

a(n) = A051523(n, 2)*(-1)^(n-1).

E.g.f.: -log(1-x)/(1-x)^10.

a(n) = n!*Sum_{k=0..n-1}((-1)^k*binomial(-10,k)/(n-k)), for n>=1. - Milan Janjic, Dec 14 2008

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

Mathematica

f[n_] := n!*Sum[(-1)^k*Binomial[-10, k]/(n - k), {k, 0, n - 1}]; Array[f, 17, 0]
Range[0, 16]! CoefficientList[ Series[-Log[(1 - x)]/(1 - x)^10, {x, 0, 16}], x]
(* Or, using elementary symmetric functions: *)
f[k_] := k + 9; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}]
(* Clark Kimberling, Dec 29 2011 *)

Crossrefs

Cf. A049398 (first unsigned column).

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

Sequence in context: A192093 A006105 A167032 * A302308 A303039 A302876

Adjacent sequences: A051561 A051562 A051563 * A051565 A051566 A051567

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved