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A022916 Multinomial coefficient n!/([n/3]![(n+1)/3]![(n+2)/3]!). 5
1, 1, 2, 6, 12, 30, 90, 210, 560, 1680, 4200, 11550, 34650, 90090, 252252, 756756, 2018016, 5717712, 17153136, 46558512, 133024320, 399072960, 1097450640, 3155170590, 9465511770, 26293088250, 75957810500, 227873431500 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of permutation patterns modulo 3. This matches the multinomial formula. - Olivier Gérard, Feb 25 2011

Also the number of permutations of n elements where p(k-3) < p(k) for all k. - Joerg Arndt, Jul 23 2011

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

FORMULA

Recurrence: (n+1)*(n+2)*(3*n+1)*a(n) = 3*(3*n^2 + 3*n + 2)*a(n-1) + 27*(n-1)*(n+2)*a(n-2) + 27*(n-2)*(n-1)*(3*n+4)*a(n-3). - Vaclav Kotesovec, Feb 26 2014

a(n) ~ 3^(n+3/2) / (2*Pi*n). - Vaclav Kotesovec, Feb 26 2014

EXAMPLE

Starting from n=4, several permutations have the same pattern. Both (3,1,4,2) and (3,4,1,2) have pattern (0, 1, 1, 2) modulo 3.

MATHEMATICA

Table[ n!/(Quotient[n, 3]!*Quotient[n + 1, 3]!*Quotient[n + 2, 3]!), {n, 0, 30}]

Table[n!/Times@@(Floor/@((n+{0, 1, 2})/3)!), {n, 0, 30}] (* Harvey P. Dale, Jul 13 2012 *)

Table[Multinomial[Floor[n/3], Floor[(n+1)/3], Floor[(n+2)/3]], {n, 0, 30}] (* Jean-François Alcover, Jun 24 2015 *)

PROG

(PARI) a(n)=n!/((n\3)!*((n+1)\3)!*((n+2)\3)!)

(PARI) {a(n)= if(n<0, 0, n!/(n\3)!/((n+1)\3)!/((n+2)\3)!)} /* Michael Somos, Jun 20 2007 */

CROSSREFS

A006480(n) = a(3*n).

Cf. A001405 (permutation patterns mod 2).

Cf. A022917 (permutation patterns mod 4).

Sequence in context: A291518 A291445 A320664 * A073949 A080372 A163087

Adjacent sequences:  A022913 A022914 A022915 * A022917 A022918 A022919

KEYWORD

nonn,easy,nice

AUTHOR

Clark Kimberling, Jun 14 1998

EXTENSIONS

Corrected by Michael Somos, Jun 20 2007

STATUS

approved

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Last modified December 19 02:33 EST 2018. Contains 318245 sequences. (Running on oeis4.)