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A193638
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Number of permutations of the multiset {1,1,1,2,2,2,3,3,3,...,n,n,n} with no two consecutive terms equal.
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6
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1, 0, 2, 174, 41304, 19606320, 16438575600, 22278418248240, 45718006789687680, 135143407245840698880, 553269523327347306412800, 3039044104423605600086688000, 21819823367694505460651694873600, 200345011881335747639978525387827200
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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EXAMPLE
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a(2) = 2 because there are two permutations of {1,1,1,2,2,2} avoiding equal consecutive terms: 121212 and 212121.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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]]) >;
(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))
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CROSSREFS
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Cf. A114938 = similar, with two copies instead of three.
Cf. A193624 = arrangements of triples with no adjacent siblings.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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