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A001716 Generalized Stirling numbers.
(Formerly M4651 N1990)
16
1, 9, 74, 638, 5944, 60216, 662640, 7893840, 101378880, 1397759040, 20606463360, 323626665600, 5395972377600, 95218662067200, 1773217155225600, 34758188233574400, 715437948072960000, 15429680577561600000, 347968129734973440000, 8190600438533990400000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The asymptotic expansion of the higher order exponential integral E(x,m=2,n=4) ~ exp(-x)/x^2*(1 - 9/x + 74/x^2 - 638/x^3 + 5944/x^4 - 60216/x^5 + 662640/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.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..100

FORMULA

a(n) = sum((-1)^(n+k)*(k+1)*4^k*stirling1(n+1, k+1), k=0..n). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

a(n)=n!*sum((-1)^k*binomial(-4,k)/(n-k),k=0..n-1). [From Milan Janjic, Dec 14 2008]

a(n)=n!*[3]h(n), where [k]h(n) denotes the k-th successive summation of the harmonic numbers from 0 to n. With offset 1. [From Gary Detlefs Jan 04 2011]

a(n)=(n+1)!*sum((-1)^k*binomial(-4,k)/(n+1-k),k=0..n). [From Gary Detlefs, Jul 16 2011]

a(n)=(n+4)!*sum(1/(k+3), k=1..n+1)/6. [From Gary Detlefs, Sep 14 2011]

E.g.f. (for offset 1): 1/(1-x)^4 * log(1/(1-x)). - Vaclav Kotesovec, Jan 19 2014

MATHEMATICA

f[k_] := k + 3; 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 *)

Rest[CoefficientList[Series[(1-x)^(-4)*Log[1/(1-x)], {x, 0, 20}], x]*Range[0, 20]!] (* Vaclav Kotesovec, Jan 19 2014 *)

CROSSREFS

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.

Sequence in context: A037533 A178827 A190984 * A231910 A028991 A102094

Adjacent sequences:  A001713 A001714 A001715 * A001717 A001718 A001719

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

STATUS

approved

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Last modified October 30 13:49 EDT 2014. Contains 248804 sequences.