A000482 Unsigned Stirling numbers of first kind s(n,5).

1, 15, 175, 1960, 22449, 269325, 3416930, 45995730, 657206836, 9957703756, 159721605680, 2706813345600, 48366009233424, 909299905844112, 17950712280921504, 371384787345228000, 8037811822645051776, 181664979520697076096, 4280722865357147142912, 105005310755917452984576

Offset: 5

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of permutations of n elements with exactly 5 cycles.

Let P(n-1,X) = (X+1)(X+2)(X+3)...(X+n-1); then a(n) is the coefficient of X^4; or a(n) = P''''(n-1,0)/4! - Benoit Cloitre, May 09 2002 [Edited by Petros Hadjicostas, Jun 29 2020 to agree with the offset of 5]

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

Examples

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

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].

Crossrefs

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

Programs

Mathematica

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

PARI

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

Sage

[stirling_number1(i,5) for i in range(5,22)] # Zerinvary Lajos, Jun 27 2008

Formula

E.g.f.: (-log(1-x))^5/5!. [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]

a(n) = 5*(n-3)*a(n-1) - 5*(2*n^2 - 14*n + 25)*a(n-2) + 5*(n-4)*(2*n^2 - 16*n + 33)*a(n-3) - (5*n^4 - 90*n^3 + 610*n^2 - 1845*n + 2101)*a(n-4) + (n-5)^5*a(n-5). - Vaclav Kotesovec, Feb 24 2025