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A001810
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a(n) = n!*n*(n-1)*(n-2)/36.
(Formerly M5019 N2163)
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6
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0, 0, 0, 1, 16, 200, 2400, 29400, 376320, 5080320, 72576000, 1097712000, 17563392000, 296821324800, 5288816332800, 99165306240000, 1952793722880000, 40311241850880000, 870722823979008000, 19645683716026368000, 462251381553561600000, 11325158848062259200000
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OFFSET
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0,5
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COMMENTS
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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".
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REFERENCES
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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).
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LINKS
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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].
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FORMULA
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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
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)
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EXAMPLE
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G.f. = x^3 + 16*x^4 + 200*x^5 + 2400*x^6 + 29400*x^7 + 376320*x^8 + ...
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MAPLE
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[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
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MATHEMATICA
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Table[n! n*(n-1)*(n-2)/36, {n, 0, 20}] (* T. D. Noe, Aug 10 2012 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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