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A000454
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Unsigned Stirling numbers of first kind s(n,4).
(Formerly M4730 N2022)
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10
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1, 10, 85, 735, 6769, 67284, 723680, 8409500, 105258076, 1414014888, 20313753096, 310989260400, 5056995703824, 87077748875904, 1583313975727488, 30321254007719424, 610116075740491776
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OFFSET
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4,2
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COMMENTS
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Number of permutations of n elements with exactly 4 cycles.
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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.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.
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) [From Shanzhen Gao, Sep 14 2010]
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LINKS
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T. D. Noe, Table of n, a(n) for n=4..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].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 33
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FORMULA
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Let P(n+2,X)=(X+1)(X+2)(X+3)...(X+n+2); then a(n) is the coefficient of X^3; or a(n)=P'''(n+2,0)/3! - Benoit Cloitre, May 09 2002
E.g.f.: (-log(1-x))^4/4! or (1-x)^-1 * (-log(1-x))^3. [Corrected by Joerg Arndt, Oct 05 2009]
a(n) is coefficient of x^(n+4) in (-log(1-x))^4, multiplied by (n+4)!/4!
[h(n-1, 1)^3-3*h(n-1, 1)*h(n-1, 2)+2*h(n-1, 3)]*(n-1)!/3!, h(n, r)=sum(1/i^r, i=1..n).
a(n) = det(S(i+4,j+3)|, 1 <= i,j <= n-4), 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))^4 = x^4 + 2*x^5 + 17/6*x^6 + 7/2*x^7 + ...
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MATHEMATICA
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Abs[StirlingS1[Range[4, 20], 4]] (* From Harvey P. Dale, Aug 26 2011 *)
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PROG
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(PARI) for(n=3, 50, print1(polcoeff(prod(i=1, n, x+i), 3, x), ", "))
sage: [stirling_number1(i, 4) for i in xrange(4, 22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008
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CROSSREFS
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Cf. A000254, A000399, A000482, A001233, A008275 (Stirling1 triangle).
Sequence in context: A081903 A144639 A038235 * A145146 A184122 A163412
Adjacent sequences: A000451 A000452 A000453 * A000455 A000456 A000457
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 18 2000
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STATUS
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approved
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