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A116219 If X_1,...,X_n is a partition of a 3n-set X into 3-blocks then a(n) is equal to the number of permutations f of X such that f(X_i) <> X_i, (i=1,...,n).

%I A116219 #17 Jun 22 2025 00:17:14
%S A116219 0,684,350352,470444112,1293433432704,6355554535465920,
%T A116219 50823027472983319296,618002474327361540442368,
%U A116219 10855431334634213344062394368,264600531449039456516679858441216,8665832467934840277899318819803484160,371368757645100314808527266212241861300224
%N A116219 If X_1,...,X_n is a partition of a 3n-set X into 3-blocks then a(n) is equal to the number of permutations f of X such that f(X_i) <> X_i, (i=1,...,n).
%H A116219 G. C. Greubel, <a href="/A116219/b116219.txt">Table of n, a(n) for n = 1..149</a>
%H A116219 Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>
%F A116219 a(n) = Sum_{j=0..n} (-6)^j*binomial(n,j)*(3*n-3*j)!.
%p A116219 a:=n->sum((-6)^i*binomial(n,i)*(3*n-3*i)!,i=0..n).
%t A116219 Table[Sum[(-6)^i*Binomial[n, i]*(3*n - 3*i)!, {i, 0, n}], {n, 1, 20}] (* _G. C. Greubel_, May 11 2019 *)
%o A116219 (PARI) {a(n) = sum(j=0,n, (-6)^j*binomial(n,j)*(3*(n-j))!)}; \\ _G. C. Greubel_, May 11 2019
%o A116219 (Magma) [(&+[(-6)^j*Binomial(n, j)*Factorial(3*n-3*j): j in [0..n]]): n in [1..20]]; // _G. C. Greubel_, May 11 2019
%o A116219 (Sage) [sum((-6)^j*binomial(n,j)*factorial(3*n-3*j) for j in (0..n)) for n in (1..20)] # _G. C. Greubel_, May 11 2019
%o A116219 (GAP) List([1..20], n-> Sum([0..n], j-> (-6)^j*Binomial(n,j)* Factorial(3*n-3*j))); # _G. C. Greubel_, May 11 2019
%Y A116219 Cf. A116218, A116220, A116221, A127888.
%K A116219 nonn
%O A116219 1,2
%A A116219 _Milan Janjic_, Apr 09 2007