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A000482
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Unsigned Stirling numbers of first kind s(n,5).
(Formerly M4983 N2142)
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19
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1, 15, 175, 1960, 22449, 269325, 3416930, 45995730, 657206836, 9957703756, 159721605680, 2706813345600, 48366009233424, 909299905844112, 17950712280921504, 371384787345228000, 8037811822645051776, 181664979520697076096, 4280722865357147142912, 105005310755917452984576
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OFFSET
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5,2
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COMMENTS
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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
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REFERENCES
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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]
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LINKS
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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].
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FORMULA
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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]
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EXAMPLE
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(-log(1-x))^5 = x^5 + 5/2*x^6 + 25/6*x^7 + 35/6*x^8 + ...
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MATHEMATICA
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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