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A116221 If X_1,...,X_n is a partition of a 5n-set X into 5-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 A116221 #20 Jun 22 2025 00:17:24
%S A116221 0,3614400,1306371456000,2432274637386240000,
%T A116221 15509750490368582860800000,265241692266421512138485760000000,
%U A116221 10332925158674345473855915900600320000000,815905363532798455769292988741440720076800000000,119621339682330952236606797649198078512534126592000000000
%N A116221 If X_1,...,X_n is a partition of a 5n-set X into 5-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 A116221 G. C. Greubel, <a href="/A116221/b116221.txt">Table of n, a(n) for n = 1..89</a>
%H A116221 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 A116221 a(n) = Sum_{j=0..n} (-120)^j*binomial(n,j)*(5*n-5*k)!.
%p A116221 a:=n->sum((-120)^i*binomial(n,i)*(5*n-5*i)!,i=0..n).
%t A116221 Table[Sum[(-5!)^j*Binomial[n, j]*(5*n-5*j)!, {j,0,n}], {n,1,20}] (* _G. C. Greubel_, May 11 2019 *)
%o A116221 (PARI) {a(n) = sum(j=0,n, (-120)^j*binomial(n,j)*(5*(n-j))!)}; \\ _G. C. Greubel_, May 11 2019
%o A116221 (Magma) [(&+[(-120)^j*Binomial(n, j)*Factorial(5*n-5*j): j in [0..n]]): n in [1..20]]; // _G. C. Greubel_, May 11 2019
%o A116221 (Sage) [sum((-120)^j*binomial(n,j)*factorial(5*n-5*j) for j in (0..n)) for n in (1..20)] # _G. C. Greubel_, May 11 2019
%o A116221 (GAP) List([1..20], n-> Sum([0..n], j-> (-120)^j*Binomial(n,j)* Factorial(5*n-5*j))); # _G. C. Greubel_, May 11 2019
%Y A116221 Cf. A116218, A116219, A116220.
%K A116221 nonn
%O A116221 1,2
%A A116221 _Milan Janjic_, Apr 09 2007