OFFSET
0,5
COMMENTS
a(n) is the total number of 3-2-1 patterns in all permutations on [n]. This is because there are n! permutations, binomial(n,3) triples in each one and the probability that a given triple of entries in a random permutation form a 3-2-1 pattern (or any other specified pattern of length 3) is 1/6. - David Callan, Oct 26 2006
Old name was "Coefficients of Laguerre polynomials".
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. 799.
Cornelius Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 519.
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 = 0..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].
Cornelius Lanczos, Applied Analysis. (Annotated scans of selected pages)
FORMULA
a(n) = -A021009(n, 3), n >= 0. a(n) = ((n!/3!)^2)/(n-3)!, n >= 3.
E.g.f.: x^3/(3!*(1-x)^4).
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n) = (-1)^(n-1) * f(n,3,-4), (n >= 3). - Milan Janjic, Mar 01 2009
a(n) = Sum_{k>0} k * A263771(n,k). - Alois P. Heinz, Oct 27 2015
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=3} 1/a(n) = 9*(2*e + gamma - Ei(1) - 4), where e = A001113, gamma = A001620, and Ei(1) = A091725.
Sum_{n>=3} (-1)^(n+1)/a(n) = 63*(gamma - Ei(-1)) - 36*(1/e + 1), where Ei(-1) = -A099285. (End)
EXAMPLE
G.f. = x^3 + 16*x^4 + 200*x^5 + 2400*x^6 + 29400*x^7 + 376320*x^8 + ...
MAPLE
[seq(n!*n*(n-1)*(n-2)/36, n=0..30)];
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+1), right=Set(U, card<r), U=Sequence(Z, card>=1)}, labeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=0..20) ; # Zerinvary Lajos, Feb 07 2008
MATHEMATICA
Table[n! n*(n-1)*(n-2)/36, {n, 0, 20}] (* T. D. Noe, Aug 10 2012 *)
PROG
(Sage) [factorial(m) * binomial(m, 3) / 6 for m in range(22)] # Zerinvary Lajos, Jul 05 2008
(PARI) for(n=0, 20, print1(n!*n*(n-1)*(n-2)/36, ", ")) \\ G. C. Greubel, May 16 2018
(Magma) [Factorial(n)*n*(n-1)*(n-2)/36: n in [0..20]]; // G. C. Greubel, May 16 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, Apr 12 2014
STATUS
approved