A166734
Determinant of the adjacency matrix of the n-cube graph Q_n.
Original entry on oeis.org
0, -1, 0, 9, 0, 1476225, 0, 32724184981958652351324462890625, 0
Offset: 0
a(3) = (3^1)*(1^3)*((-1)^3)*((-3)^1) = 3*1*(-1)*(-3) = 9.
- N. Biggs, Algebraic Graph Theory, Cambridge Univ. Press 1974, p. 145.
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a:= n-> mul((n-2*j)^binomial(n, j), j=0..n):
seq(a(n), n=0..9); # Alois P. Heinz, Jun 12 2022
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a(n) = prod(j=0, n, (n-2*j)^binomial(n,j)); \\ Michel Marcus, Feb 08 2020
A229334
Product of numbers of elements of nonempty subsets of divisors of n.
Original entry on oeis.org
1, 2, 2, 24, 2, 20736, 2, 20736, 24, 20736, 2, 11501279977342425366528000000, 2, 20736, 20736, 309586821120, 2, 11501279977342425366528000000, 2, 11501279977342425366528000000, 20736, 20736, 2
Offset: 1
For n = 4; divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; product of numbers of elements of subsets = 1*1*1*2*2*2*3 = 24.
For n = 4; tau(4) = 3; a(4) = [1^(3!/((3-1)!*1!))] * [2^(3!/((3-2)!*2!))] * [3^(3!/((3-3)!*3!))] = 1^3 * 2^3 * 3^1 = 24.
A365447
Number of nonempty choice functions on a set of n alternatives.
Original entry on oeis.org
1, 3, 189, 26254935, 392654823152462915625, 28032331438680332717218961936012273854096893310546875
Offset: 1
a(1) = 1 since 2^{1} = {{}, {1}} and there exists only one function f:2^{1}/{{}}->2^{1}/{{}} satisfying f(X) is a nonempty subset of any nonempty X in 2^A, i.e., f({1})={1}.
- F. Aleskerov, D. Bouyssou, and B. Monjardet, Utility, Maximization, Choice and Preference, Springer, 2007, pp. 28-31.
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a[n_] := Product[(2^k - 1)^Binomial[n, k], {k, 1, n}]; Array[a, 6] (* Amiram Eldar, Oct 03 2023 *)
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