A193638 Number of permutations of the multiset {1,1,1,2,2,2,3,3,3,...,n,n,n} with no two consecutive terms equal.

1, 0, 2, 174, 41304, 19606320, 16438575600, 22278418248240, 45718006789687680, 135143407245840698880, 553269523327347306412800, 3039044104423605600086688000, 21819823367694505460651694873600, 200345011881335747639978525387827200

Offset

0,3

Links

Andrew Woods, Table of n, a(n) for n = 0..101

H. Eriksson and A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.

Formula

a(n) = A190826(n) * n! for n >= 1.

a(n) = A193624(n)/6^n.

a(n) = Sum_{s+t+u=n} ((-1)^t*multinomial(n;s,t,u)*(3s+2t+u)!/(3!)^s. - Alexis Martin, Nov 16 2017.

a(n) = (1/6^n) * Sum_{j=0..2*n} Sum_{k=ceiling(j/2)..n} (n+j)! * binomial(2*k, j) * binomial(n, k) * (-3)^(n+k-j). - Tani Akinari, Sep 23 2012

a(n) = n*( (3*n-1)*(3*n^2-5*n+4)*a(n-1) +2*(n-1)*(6*n^2-9*n-1)*a(n-2) -4*n*(n-1)*(n-2)*a(n-3) )/(2*n-2). - Alois P. Heinz, Jun 05 2013

Example

a(2) = 2 because there are two permutations of {1,1,1,2,2,2} avoiding equal consecutive terms: 121212 and 212121.

Maple

a:= proc(n) option remember; `if`(n<3, (n-1)*(3*n-2)/2,
    n*((3*n-1)*(3*n^2-5*n+4) *a(n-1) +2*(n-1)*(6*n^2-9*n-1) *a(n-2)
    -4*n*(n-1)*(n-2) *a(n-3))/(2*n-2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 05 2013

Mathematica

a[n_]:= (1/6^n)*Sum[(n+j)!*Binomial[n, k]*Binomial[2k, j]*(-3)^(n+k-j), {j, 0, 2n}, {k, Ceiling[j/2], n}]; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Jul 22 2017, after Tani Akinari *)

Prog

(Maxima) a(n):= (1/6^n)*sum((n+j)!*sum(binomial(n, k)*binomial(2*k, j)* (-3)^(n+k-j), k, ceiling(j/2), n), j, 0, 2*n); /* Tani Akinari, Sep 23 2012 */
(Python)
from sympy.core.cache import cacheit
@cacheit
def a(n): return (n-1)*(3*n-2)//2 if n<3 else n*((3*n-1)*(3*n**2 - 5*n + 4)*a(n-1) + 2*(n-1)*(6*n**2 -9*n-1)*a(n-2) - 4*n*(n-1)*(n-2)*a(n- 3))//(2*n-2)
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 22 2017, formula after Maple code
(Magma)
B:=Binomial;
f:= func< n, j | (&+[B(n, k)*B(2*k, j)*(-3)^(k-j): k in [Ceiling(j/2)..n]]) >;
A193638:= func< n | (-1/2)^n*(&+[Factorial(n+j)*f(n, j): j in [0..2*n]]) >;
[A193638(n): n in [0..30]]; // G. C. Greubel, Sep 22 2023
(SageMath)
b=binomial;
def f(j, n): return sum(b(n, k)*b(2*k, j)*(-3)^(k-j) for k in range((j//2), n+1))
def A193638(n): return (-1/2)^n*sum(factorial(n+j)*f(j, n) for j in range(2*n+1))
[A193638(n) for n in range(31)] # G. C. Greubel, Sep 22 2023

Crossrefs

Cf. A114938 = similar, with two copies instead of three.

Cf. A193624 = arrangements of triples with no adjacent siblings.

Cf. A190826.

Sequence in context: A281958 A172231 A360478 * A215123 A321634 A219724

Adjacent sequences: A193635 A193636 A193637 * A193639 A193640 A193641

Keyword

nonn

Author

Andrew Woods, Aug 01 2011

Status

approved