A000399 Unsigned Stirling numbers of first kind s(n,3).

1, 6, 35, 225, 1624, 13132, 118124, 1172700, 12753576, 150917976, 1931559552, 26596717056, 392156797824, 6165817614720, 102992244837120, 1821602444624640, 34012249593822720, 668609730341153280, 13803759753640704000

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Number of permutations of n elements with exactly 3 cycles.

The asymptotic expansion of the higher order exponential integral E(x,m=3,n=1) ~ exp(-x)/x^3*(1 - 6/x + 35/x^2 - 225/x^3 + 1624/x^4 - 13132/x^5 + ...) leads to the sequence given above. See A163931 and A163932 for more information. - Johannes W. Meijer, Oct 20 2009

Examples

			(-log(1-x))^3 = x^3 + 3/2*x^4 + 7/4*x^5 + 15/8*x^6 + ...
		

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.
• Shanzhen Gao, Permutations with Restricted Structure (in preparation). - Shanzhen Gao, Sep 14 2010
• 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).

Links

• T. D. Noe and Robert Israel, Table of n, a(n) for n = 3..412 (3..100 from T. D. Noe)
• 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 32.
• Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, arXiv:1201.1323 [math.CO], 2012.
• Sergey Kitaev and Jeffrey Remmel, Quadrant Marked Mesh Patterns, J. Int. Seq. 15 (2012), #12.4.7.

Crossrefs

Cf. A000254, A000454, A000482, A001233, A001234, A008275, A243569, A243570.

Programs

Magma

A000399:=func< n | Abs(StirlingFirst(n, 3)) >; [ A000399(n): n in [3..25] ]; // Klaus Brockhaus, Jan 14 2011

Maple

seq(abs(Stirling1(n,3)),n=3..30); # Robert Israel, Jul 05 2015

Mathematica

a=Log[1/(1-x)];Range[0,20]! CoefficientList[Series[a^3/3!,{x,0,20}],x]
f[n_] := Abs@ StirlingS1[n, 3]; Array[f, 19, 3]
Abs[StirlingS1[Range[3,30],3]] (* Harvey P. Dale, Jun 23 2014 *)
f[n_] := Gamma[n]*(HarmonicNumber[n - 1]^2 + Zeta[2, n] - Zeta[2])/2; Array[f, 19, 3] (* Robert G. Wilson v, Jul 05 2015 *)

MuPAD

f := proc(n) option remember; begin n^3*f(n-3)-(3*n^2+3*n+1)*f(n-2)+3*(n+1)*f(n-1) end_proc: f(0) := 1: f(1) := 6: f(2) := 35:

PARI

for(n=2,50,print1(polcoeff(prod(i=1,n,x+i),2,x),","))

Sage

[stirling_number1(i+2,3) for i in range(1,22)] # Zerinvary Lajos, Jun 27 2008

Formula

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

E.g.f.: -log(1-x)^3/3!.

a(n) is the coefficient of x^(n+3) in (-log(1-x))^3, multiplied by (n+3)!/6.

a(n) = ((Sum_{i=1..n-1} 1/i)^2 - Sum_{i=1..n-1} 1/i^2)*(n-1)!/2 for n >= 3. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 18 2000

a(n) = det(|S(i+3,j+2)|, 1 <= i,j <= n-3), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 06 2013

a(n) = Gamma(n)*(HarmonicNumber(n-1)^2 + Zeta(2,n) - Zeta(2))/2. - Gerry Martens, Jul 05 2015

From Petros Hadjicostas, Jun 28 2020: (Start)

a(n) = (n-3)! + (2*n-3)*a(n-1) - (n-2)^2*a(n-2) for n >= 5.

a(n) = 3*(n-2)*a(n-1) - (3*n^2-15*n+19)*a(n-2) + (n-3)^3*a(n-3) for n >= 6. (End)