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A001233 Unsigned Stirling numbers of first kind s(n,6).
(Formerly M5114 N2216)
12
1, 21, 322, 4536, 63273, 902055, 13339535, 206070150, 3336118786, 56663366760, 1009672107080, 18861567058880, 369012649234384, 7551527592063024, 161429736530118960, 3599979517947607200, 83637381699544802976, 2021687376910682741568, 50779532534302850198976, 1323714091579185857760000 (list; graph; refs; listen; history; text; internal format)
OFFSET

6,2

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]

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].

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]

EXAMPLE

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

MATHEMATICA

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

PROG

(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

CROSSREFS

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

Sequence in context: A036737 A141267 A016262 * A145148 A214099 A340097

Adjacent sequences:  A001230 A001231 A001232 * A001234 A001235 A001236

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified March 7 17:11 EST 2021. Contains 341897 sequences. (Running on oeis4.)