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

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, December 1972 [alternative scanned copy].

Index entries for sequences related to Laguerre polynomials

Zerinvary Lajos, Sage Notebooks

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).

[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: A081679 A125451 A129333 this_sequence A016165 A016217 A055758

Adjacent sequences: A001807 A001808 A001809 this_sequence A001811 A001812 A001813

nonn

njas