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A000482 Unsigned Stirling numbers of first kind s(n,5).
(Formerly M4983 N2142)
13
1, 15, 175, 1960, 22449, 269325, 3416930, 45995730, 657206836, 9957703756, 159721605680, 2706813345600, 48366009233424, 909299905844112, 17950712280921504, 371384787345228000, 8037811822645051776, 181664979520697076096, 4280722865357147142912, 105005310755917452984576 (list; graph; refs; listen; history; text; internal format)
OFFSET

5,2

COMMENTS

Number of permutations of n elements with exactly 5 cycles.

Let P(n+3,X)=(X+1)(X+2)(X+3)...(X+n+3); then a(n) is the coefficient of X^4; or a(n)=P''''(n+3,0)/4! - Benoit Cloitre, May 09 2002

The asymptotic expansion of the higher order exponential integral E(x,m=5,n=1) ~ exp(-x)/x^5*(1 - 15/x + 175/x^2 - 1960/x^3 + 22449/x^4 - ...) leads to the sequence given above. See A163931 for E(x,m,n) information and A163932 for a Maple procedure for the asymptotic expansion. - Johannes W. Meijer, Oct 20 2009

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.

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).

Shanzhen Gao, Permutations with Restricted Structure (in preparation) [Shanzhen Gao, Sep 14 2010]

LINKS

T. D. Noe, Table of n, a(n) for n=5..100

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

FORMULA

E.g.f. (-log(1-x))^5/5! or (1-x)^-1 * (-log(1-x))^4 [Corrected by Joerg Arndt, Oct 05 2009]

a(n) is coefficient of x^(n+5) in (-log(1-x))^5, multiplied by (n+5)!/5!

a(n) = det(S(i+5,j+4)|, 1 <= i,j <= n-5), where S(n,k) are Stirling numbers of the second kind. [Mircea Merca, Apr 06 2013]

EXAMPLE

(-log(1-x))^5 = x^5 + 5/2*x^6 + 25/6*x^7 + 35/6*x^8 + ...

MATHEMATICA

Abs[StirlingS1[Range[5, 30], 5]] (* Harvey P. Dale, May 26 2014 *)

PROG

(PARI) for(n=4, 50, print1(polcoeff(prod(i=1, n, x+i), 4, x), ", "))

sage: [stirling_number1(i, 5) for i in xrange(5, 22)] - Zerinvary Lajos, Jun 27 2008

CROSSREFS

Cf. A000254, A000399, A000454, A001233, A001234, A243569, A243570, A008275.

Sequence in context: A036083 A051588 A016164 * A145147 A069379 A184285

Adjacent sequences:  A000479 A000480 A000481 * A000483 A000484 A000485

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 25 15:14 EDT 2017. Contains 284082 sequences.