4,2

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

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.

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, December 1972 [alternative scanned copy].

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 33

E.g.f.: (-log(1-x))^4 or (1-x)^-1 * (-log(1-x))^3

a(n) is coefficient of x^(n+4) in (-log(1-x))^4, multiplied by (n+4)!/4!

[h(n-1, 1)^3-3*h(n-1, 1)*h(n-1, 2)+2*h(n-1, 3)]*(n-1)!/3!, h(n, r)=sum(1/i^r, i=1..n).

(-log(1-x))^4 = x^4 + 2*x^5 + 17/6*x^6 + 7/2*x^7 + ...

(PARI) for(n=3, 50, print1(polcoeff(prod(i=1, n, x+i), 3, x), ", "))

sage: [stirling_number1(i, 4) for i in xrange(4, 22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008

Cf. A000254, A000399, A000482, A001233, A008275 (Stirling1 triangle).

Adjacent sequences: A000451 A000452 A000453 this_sequence A000455 A000456 A000457

Sequence in context: A014341 A081903 A038235 this_sequence A121115 A114648 A136864

nonn

njas

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