|
|
A000454
|
|
Unsigned Stirling numbers of first kind s(n,4).
(Formerly M4730 N2022)
|
|
19
|
|
|
1, 10, 85, 735, 6769, 67284, 723680, 8409500, 105258076, 1414014888, 20313753096, 310989260400, 5056995703824, 87077748875904, 1583313975727488, 30321254007719424, 610116075740491776
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
4,2
|
|
COMMENTS
|
Number of permutations of n elements with exactly 4 cycles.
The asymptotic expansion of the higher order exponential integral E(x, m=4, n=1) ~ exp(-x)/x^4*(1 - 10/x + 85/x^2 - 735/x^3 + 6769/x^4 - ...) leads to the sequence given above. See A163931 and A163932 for more information. - Johannes W. Meijer, Jun 11 2016
|
|
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.
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Shanzhen Gao, Permutations with Restricted Structure (in preparation) [From Shanzhen Gao, Sep 14 2010] [Apparently unpublished as of June 2016]
|
|
LINKS
|
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
|
Let P(n-1,X) = (X+1)(X+2)(X+3)...(X+n-1); then a(n) is the coefficient of X^3; or a(n) = P'''(n-1,0)/3!. - Benoit Cloitre, May 09 2002 [Edited by Petros Hadjicostas, Jun 29 2020 to agree with the offset 4]
E.g.f.: (-log(1-x))^4/4!. [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!.
a(n) = (h(n-1, 1)^3 - 3*h(n-1, 1)*h(n-1, 2) + 2*h(n-1, 3))*(n-1)!/3!, where h(n, r) = Sum_{i=1..n} 1/i^r. - Klaus Strassburger, 2000
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
a(n) = y(n)*n!/24, where y(0) = y(1) = y(2) = y(3) = 0, y(4) = 1 and n^4*y(n) + (-1-5*n-10*n^2-10*n^3-4*n^4)*y(n+1) + (1+n)*(2+n)*(7+12*n+6*n^2)*y(n+2) - 2*(1+n)*(2+n)*(3+n)*(3+2*n)*y(3+n) + (1+n)*(2+n)*(3+n)*(4+n)*y(n+4) = 0. - Benedict W. J. Irwin, Jul 12 2016
a(n) = 2*(2*n - 5)*a(n-1) - (6*n^2 - 36*n + 55)*a(n-2) + (2*n - 7)*(2*n^2 - 14*n + 25)*a(n-3) - (n-4)^4*a(n-4).
a(n) ~ n! * (log(n))^3 / (6*n) * (1 + 3*gamma/log(n) + (3*gamma^2 - Pi^2/2)/ (log(n))^2), where gamma is the Euler-Mascheroni constant A001620. (End)
a(n) = A000399(n-1) + (n-1)*a(n-1) for n >= 1 (assuming a(n) = 0 for n = 0..3).
a(n) = A103719(n-4) + (n-2)*a(n-1) for n >= 4.
a(n) = A000254(n-3) + (2*n-3)*a(n-1) - (n-2)^2*a(n-2) for n >= 3.
a(n) = (n-4)! + 3*(n-2)*a(n-1) - (3*n^2-15*n+19)*a(n-2) + (n-3)^3*a(n-3) for n >= 4. (End)
|
|
EXAMPLE
|
(-log(1-x))^4 = x^4 + 2*x^5 + (17/6)*x^6 + (7/2)*x^7 + ...
|
|
MATHEMATICA
|
|
|
PROG
|
(PARI) for(n=3, 50, print1(polcoeff(prod(i=1, n, x+i), 3, x), ", "))
(Sage) [stirling_number1(i, 4) for i in range(4, 22)] # Zerinvary Lajos, Jun 27 2008
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 18 2000
|
|
STATUS
|
approved
|
|
|
|