0,5

a(n) is the total number of 3-2-1 patterns in all permutations on [n]. This is because there are n! permutations, binom(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 (callan(AT)stat.wisc.edu), Oct 26 2006

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 519.

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].

Index entries for sequences related to Laguerre polynomials

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(sum(binomial(k,j)*stirling1(n,k)*stirling2(j,i)*x^(k-j),j=i..k),k=i..n) then a(n)=(-1)^(n-1)*f(n,3,-4), (n>=3). [From Milan R. Janjic (agnus(AT)blic.net), Mar 01 2009]

[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 (zerinvarylajos(AT)yahoo.com), Feb 07 2008

sage: [factorial(m)*binomial(m, 3)/6 for m in xrange (0, 22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 05 2008

Cf. A053495.

Sequence in context: A125451 A154348 A129333 this_sequence A016165 A144632 A161729

Adjacent sequences: A001807 A001808 A001809 this_sequence A001811 A001812 A001813

nonn

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