|
| |
|
|
A001711
|
|
Generalized Stirling numbers.
(Formerly M4429 N1873)
|
|
20
| |
|
|
1, 7, 47, 342, 2754, 24552, 241128, 2592720, 30334320, 383970240, 5231113920, 76349105280, 1188825724800, 19675048780800, 344937224217600, 6386713749964800, 124548748102195200, 2551797512248320000
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 20 2009: (Start)
The asymptotic expansion of the higher order exponential integral E(x,m=2,n=3) ~ exp(-x)/x^2*(1 - 7/x + 47/x^2 - 342/x^3 + 2754/x^4 - 24552/x^5 + 241128/x^6 - ... ) leads to the sequence given above. See A163931 and A028421 for more information.
(End)
For n>4, a(n) mod n = 0 for n composite, = n-3 for n prime. [From Gary Detlefs, Jul 18 2011]
|
|
|
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).
|
|
|
FORMULA
| E.g.f.: - ln(1 - x)/(1 - x)^3 if offset 1. With offset 0: diff(- ln(1 - x)/(1 - x)^3,x) = (1 - 3*ln(1 - x))/(1 - x)^4.
a(n)=sum((-1)^(n+k)*(k+1)*3^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(-3,k)/(n-k),k=0..n-1); [From Milan R. Janjic (agnus(AT)blic.net), Dec 14 2008]
a(n) = ( A000254(n+3) -3* A001710(n+3) )/2 [From Gary Detlefs, May 24 2010]
a(n)= (n+3)!/4 * (2*h(n+3)-3), where h(n)=sum(1/k,k=1..n) [From Gary Detlefs,Aug 15 2010]
a(n)=n!*[2]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+3)!*sum(1/(2*k+4), k=1..n+1). [From Gary Detlefs, Sep 14 2011]
a(n) = (n+1)!* sum(binomial(k+2,2)/(n+1-k), k=0..n). [From Gary Detlefs, Dec 01 2011]
|
|
|
MAPLE
| a:=n->sum(1/2*(n!/k), k=3..n): seq(a(n), n=3..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 22 2008
h:=n->sum(1/k, k=0..n):hs:=n->sum(h(k), k=0..n):hs2:=n->sum(hs(k), k=0..n):seq(n!*hs2(n), n=1..18); [From Gary Detlefs (gdetlefs(AT)aol.com) Jan 01 2011]
|
|
|
MATHEMATICA
| f[k_] := k + 2; 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
| 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 [From Gary Detlefs (gdetlefs(AT)aol.com) Jan 04 2011]
Sequence in context: A098405 A104092 A024187 * A088057 A108434 A093173
Adjacent sequences: A001708 A001709 A001710 * A001712 A001713 A001714
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
|
| |
|
|
