6,2

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 20 2009: (Start)

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 See A163931 for E(x,m,n) information and A163932 for a Maple procedure for the asymptotic expansion.

(End)

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).

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].

Let P(n+4,X)=(X+1)(X+2)(X+3)...(X+n+4); then a(n) is the coefficient of X^5; or a(n)=P'''''(n+4,0)/5! - Benoit Cloitre (benoit7848c(AT)orange.fr), May 09 2002

E.g.f.: (-log(1-x))^6/6! or (1-x)^-1 * (-log(1-x))^5.

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

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

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

sage: [stirling_number1(i, 6) for i in xrange(6, 22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008

Cf. A000254, A000399, A000454, A000482, A008275 (Stirling1 triangle).

Sequence in context: A036737 A141267 A016262 this_sequence A145148 A016260 A011810

Adjacent sequences: A001230 A001231 A001232 this_sequence A001234 A001235 A001236

nonn,easy,more

N. J. A. Sloane (njas(AT)research.att.com).