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A001810 a(n) = n!*n*(n-1)*(n-2)/36.
(Formerly M5019 N2163)
6
0, 0, 0, 1, 16, 200, 2400, 29400, 376320, 5080320, 72576000, 1097712000, 17563392000, 296821324800, 5288816332800, 99165306240000, 1952793722880000, 40311241850880000, 870722823979008000, 19645683716026368000, 462251381553561600000, 11325158848062259200000 (list; graph; refs; listen; history; text; internal format)
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, 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, 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.

C. 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].

C. Lanczos, Applied Analysis (Annotated scans of selected pages)

Index entries for sequences related to Laguerre polynomials

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

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 xrange (0, 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

Cf. A053495, A263771.

Sequence in context: A125451 A154348 A129333 * A016165 A282834 A144632

Adjacent sequences:  A001807 A001808 A001809 * A001811 A001812 A001813

KEYWORD

nonn,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by N. J. A. Sloane, Apr 12 2014

STATUS

approved

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Last modified May 23 07:21 EDT 2018. Contains 304455 sequences. (Running on oeis4.)