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Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed pair (j,j).
7

%I #32 Aug 26 2024 16:07:51

%S 1,0,5,74,2193,101644,6840085,630985830,76484389121,11792973495032,

%T 2254432154097861,523368281765512930,145044815855963403985,

%U 47302856057098946329284,17933275902554972391519893,7820842217155394547769452734,3887745712142302082441578104705

%N Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed pair (j,j).

%C Inverse binomial transform of A000680.

%H Alois P. Heinz, <a href="/A374980/b374980.txt">Table of n, a(n) for n = 0..238</a>

%F a(n) = Sum_{j=0..n} (-1)^j*binomial(n,j)*A000680(n-j).

%F a(n) = A116218(n)/2^n.

%F a(n) mod 2 = 1 - (n mod 2) = A059841(n).

%e a(2) = 5: 1212, 1221, 2112, 2121, 2211.

%p a:= proc(n) option remember; `if`(n<3, [1, 0, 5][n+1],

%p (n-1)*((2*n+1)*a(n-1)+(4*n-3)*a(n-2)+2*(n-2)*a(n-3)))

%p end:

%p seq(a(n), n=0..16);

%Y Cf. A000166, A000459, A000680, A059841, A116218, A375222, A375223.

%Y Column k=2 of A375694.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 05 2024