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A001810 a(n) = n!*n*(n-1)*(n-2)/36.

%I A001810 M5019 N2163 #52 Sep 08 2022 08:44:29
%S A001810 0,0,0,1,16,200,2400,29400,376320,5080320,72576000,1097712000,
%T A001810 17563392000,296821324800,5288816332800,99165306240000,
%U A001810 1952793722880000,40311241850880000,870722823979008000,19645683716026368000,462251381553561600000,11325158848062259200000
%N A001810 a(n) = n!*n*(n-1)*(n-2)/36.
%C A001810 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
%C A001810 Old name was "Coefficients of Laguerre polynomials".
%D A001810 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.
%D A001810 Cornelius Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 519.
%D A001810 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001810 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001810 T. D. Noe, <a href="/A001810/b001810.txt">Table of n, a(n) for n = 0..100</a>
%H A001810 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A001810 Cornelius Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a>. (Annotated scans of selected pages)
%H A001810 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>.
%F A001810 a(n) = -A021009(n, 3), n >= 0. a(n) = ((n!/3!)^2)/(n-3)!, n >= 3.
%F A001810 E.g.f.: x^3/(3!*(1-x)^4).
%F A001810 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
%F A001810 a(n) = Sum_{k>0} k * A263771(n,k). - _Alois P. Heinz_, Oct 27 2015
%F A001810 From _Amiram Eldar_, May 02 2022: (Start)
%F A001810 Sum_{n>=3} 1/a(n) = 9*(2*e + gamma - Ei(1) - 4), where e = A001113, gamma = A001620, and Ei(1) = A091725.
%F A001810 Sum_{n>=3} (-1)^(n+1)/a(n) = 63*(gamma - Ei(-1)) - 36*(1/e + 1), where Ei(-1) = -A099285. (End)
%e A001810 G.f. = x^3 + 16*x^4 + 200*x^5 + 2400*x^6 + 29400*x^7 + 376320*x^8 + ...
%p A001810 [seq(n!*n*(n-1)*(n-2)/36,n=0..30)];
%p A001810 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
%t A001810 Table[n! n*(n-1)*(n-2)/36, {n, 0, 20}] (* _T. D. Noe_, Aug 10 2012 *)
%o A001810 (Sage) [factorial(m) * binomial(m, 3) / 6 for m in range(22)]  # _Zerinvary Lajos_, Jul 05 2008
%o A001810 (PARI) for(n=0,20, print1(n!*n*(n-1)*(n-2)/36, ", ")) \\ _G. C. Greubel_, May 16 2018
%o A001810 (Magma) [Factorial(n)*n*(n-1)*(n-2)/36: n in [0..20]]; // _G. C. Greubel_, May 16 2018
%Y A001810 Cf. A053495, A263771.
%Y A001810 Cf. A001113, A001620, A091725, A099285.
%K A001810 nonn
%O A001810 0,5
%A A001810 _N. J. A. Sloane_
%E A001810 Edited by _N. J. A. Sloane_, Apr 12 2014