A000454 Unsigned Stirling numbers of first kind s(n,4). (Formerly M4730 N2022)

1, 10, 85, 735, 6769, 67284, 723680, 8409500, 105258076, 1414014888, 20313753096, 310989260400, 5056995703824, 87077748875904, 1583313975727488, 30321254007719424, 610116075740491776

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Shanzhen Gao, Permutations with Restricted Structure (in preparation) [From Shanzhen Gao, Sep 14 2010] [Apparently unpublished as of June 2016]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

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.

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

From Vaclav Kotesovec, Jul 12 2016: (Start)

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)

From Petros Hadjicostas, Jun 29 2020: (Start)

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

Abs[StirlingS1[Range[4, 20], 4]] (* Harvey P. Dale, Aug 26 2011 *)

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

Cf. A000254, A000399, A000482, A001233, A001234, A008275, A243569, A243570.

Sequence in context: A354390 A354318 A346946 * A347003 A346923 A145146

Adjacent sequences: A000451 A000452 A000453 * A000455 A000456 A000457

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 18 2000

Status

approved