A375694 -id:A375694 - cp's OEIS frontend

A374980 Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed pair (j,j).

1, 0, 5, 74, 2193, 101644, 6840085, 630985830, 76484389121, 11792973495032, 2254432154097861, 523368281765512930, 145044815855963403985, 47302856057098946329284, 17933275902554972391519893, 7820842217155394547769452734, 3887745712142302082441578104705

Offset: 0

Alois P. Heinz, Aug 05 2024

nonn

Inverse binomial transform of A000680.

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

Alois P. Heinz, Table of n, a(n) for n = 0..238

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

Column k=2 of A375694.

a:= proc(n) option remember; `if`(n<3, [1, 0, 5][n+1],
     (n-1)*((2*n+1)*a(n-1)+(4*n-3)*a(n-2)+2*(n-2)*a(n-3)))
    end:
seq(a(n), n=0..16);

a(n) = Sum_{j=0..n} (-1)^j*binomial(n,j)*A000680(n-j).
a(n) = A116218(n)/2^n.
a(n) mod 2 = 1 - (n mod 2) = A059841(n).

A375693 Number of multiset permutations of {{1}^n, {2}^n, ..., {n}^n} with no fixed n-tuple {j}^n.

Original entry on oeis.org

1, 0, 5, 1622, 62924817, 623302086965044, 2670169511426774520697375, 7363615066099523741730150062678073534, 18165723898797467057177720588121375861340650728031233, 53130688704554689391452744667655289291011354800478739192999936981375688

Offset: 0

Alois P. Heinz, Aug 24 2024

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..26

Main diagonal of A375694.

Cf. A034841, A059841, A306241, A375778.

a:= n-> add((-1)^(n-j)*binomial(n, j)*(n*j)!/n!^j, j=0..n):
seq(a(n), n=0..10);

a(n) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(n*j)!/n!^j.

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

cp's OEIS Frontend

A374980 Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed pair (j,j).

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