A001233 Unsigned Stirling numbers of first kind s(n,6).

1, 21, 322, 4536, 63273, 902055, 13339535, 206070150, 3336118786, 56663366760, 1009672107080, 18861567058880, 369012649234384, 7551527592063024, 161429736530118960, 3599979517947607200, 83637381699544802976, 2021687376910682741568, 50779532534302850198976, 1323714091579185857760000

Offset: 6

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The asymptotic expansion of the higher order exponential integral E(x,m=6,n=1) ~ exp(-x)/x^6*(1 - 21/x + 322/x^2 - 4536/x^3 + 63273/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

Examples

			(-log(1-x))^6 = x^6 + 3*x^7 + 23/4*x^8 + 9*x^9 + ...
		

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.
• 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).

Links

• T. D. Noe, Table of n, a(n) for n = 6..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].

Crossrefs

Cf. A000254, A000399, A000454, A000482, A001234, A008275, A243569, A243570.

Programs

Mathematica

Drop[Abs[StirlingS1[Range[30],6]],5] (* Harvey P. Dale, Sep 17 2013 *)

PARI

for(n=5,50,print1(polcoeff(prod(i=1,n,x+i),5,x),","))

Sage

[stirling_number1(i,6) for i in range(6,22)] # 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^5; or a(n) = P'''''(n-1,0)/5!. - Benoit Cloitre, May 09 2002 [Edited by Petros Hadjicostas, Jun 29 2020 to agree with the offset of 6]

E.g.f.: (-log(1-x))^6/6!.

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

a(n) = det(|S(i+6,j+5)|, 1 <= i,j <= n-6), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 06 2013

a(n) = 3*(2*n - 7)*a(n-1) - 5*(3*n^2 - 24*n + 49)*a(n-2) + 10*(2*n - 9)*(n^2 - 9*n + 21)*a(n-3) - (15*n^4 - 300*n^3 + 2265*n^2 - 7650*n + 9751)*a(n-4) + (2*n - 11)*(n^2 - 11*n + 31)*(3*n^2 - 33*n + 91)*a(n-5) - (n-6)^6*a(n-6). - Vaclav Kotesovec, Feb 24 2025