login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001706 Generalized Stirling numbers.
(Formerly M4646 N1988)
6
1, 9, 71, 580, 5104, 48860, 509004, 5753736, 70290936, 924118272, 13020978816, 195869441664, 3134328981120, 53180752331520, 953884282141440, 18037635241029120, 358689683932346880, 7483713725055744000, 163478034254755584000, 3731670622213083648000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The asymptotic expansion of the higher order exponential integral E(x,m=3,n=2) ~ exp(-x)/x^3*(1 - 9/x + 71/x^2 - 580/x^3 + 5104/x^4 - 48860/x^5+ the sequence given above. See A163931 and A163932 for more information. - Johannes W. Meijer, Oct 20 2009

a(n-1) is equal to -1 times the coefficient of x of the characteristic polynomial of the n X n matrix whose (i,j)-entry is equal to i+3 if i=j and is equal to 1 otherwise. - John M. Campbell, May 24 2011

REFERENCES

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

D. S. Mitrinovic, R. S. Mitrinovic, Tableaux d'une classe de nombres reliƩs aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]

FORMULA

E.g.f. (with offset 2): log(1 - x)^2 / (2 * (1 - x)^2).

a(n) = Sum_{k=0..n}(-1)^(n+k)*binomial(k+2, 2)*2^k*stirling1(n+2, k+2). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

a(n-1) = (1/2)*Sum_{i=0..n} binomial(n, i)*A000254(i)*A000254(n-i). - Benoit Cloitre, Mar 09 2004

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n-1) = |f(n,2,2)|, for n>=2. - Milan Janjic, Dec 21 2008

a(n) = (n+3)!*((gamma-1)*Psi(n+4)+2+gamma^2-17*gamma/6+sum(Psi(i+4)/(i+4),i = 0 .. n-1)). - Mark van Hoeij, Oct 26 2011

MATHEMATICA

Table[-Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1, #2]((((#1+3)))-1)+1&, {n, n}], x], x, 1], {n, 1, 10}] (* John M. Campbell, May 24 2011 *)

CROSSREFS

Sequence in context: A164551 A178869 A057080 * A251284 A144745 A158193

Adjacent sequences:  A001703 A001704 A001705 * A001707 A001708 A001709

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Christian G. Bower

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 29 07:21 EDT 2017. Contains 284250 sequences.