|
%I #24 Mar 14 2021 18:43:01
%S 1,1,0,1,0,0,1,1,0,0,7,7,0,0,1,1,0,0,35,105,168,280,435,435,280,168,
%T 105,35,0,0,1,1,0,0,155,1085,5208,22568,82615,247845,628680,1383096,
%U 2648919,4414865,6440560,8280720,9398115,9398115,8280720,6440560,4414865,2648919,1383096,628680,247845,82615,22568,5208,1085,155,0,0,1
%N Triangle read by rows: T(n,k) is the number of hypergraphs on n labeled vertices with k edges and all vertices having even degree, 0 <= k < 2^n.
%C Hypergraphs are graphs in which an edge is connected to a nonempty subset of vertices rather than exactly two of them. An edge is a nonempty subset of vertices.
%C Equivalently, T(n,k) is the number of subsets of {1..2^n-1} with k elements such that the bitwise-xor of the elements is zero.
%C Also the coefficients of polynomials p_{n}(x) which have the representation
%C p_{n}(x) = (x + 1)^(2*(n - 1) - 1)*q_{n - 1}(x), where q_{n}(x) are the polynomials defined in A340263, and n >= 2. - _Peter Luschny_, Jan 10 2021
%H Andrew Howroyd, <a href="/A340030/b340030.txt">Table of n, a(n) for n = 0..2046</a> (rows 0..10)
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypergraph">Hypergraph</a>.
%F T(n,k) = (binomial(2^n-1, k) + (-1)^ceiling(k/2)*(2^n-1)*binomial(2^(n-1)-1, floor(k/2)))/2^n.
%F T(n,2*k) + T(n,2*k+1) = binomial(2^n-1, k)/2^n = A281123(n,k).
%F T(n, k) = T(n, 2^n-1-k) for n >= 2.
%e Triangle begins:
%e [0] 1;
%e [1] 1, 0;
%e [2] 1, 0, 0, 1;
%e [3] 1, 0, 0, 7, 7, 0, 0, 1;
%e [4] 1, 0, 0, 35, 105, 168, 280, 435, 435, 280, 168, 105, 35, 0, 0, 1;
%o (PARI)
%o T(n,k) = {(binomial(2^n-1, k) + (-1)^((k+1)\2)*(2^n-1)*binomial(2^(n-1)-1, k\2))/2^n}
%o { for(n=0, 5, print(vector(2^n, k, T(n,k-1)))) }
%Y Rows 3..8 are A002394, A010085, A010086, A010087, A010088, A010089.
%Y Row sums are A016031(n+1).
%Y Column k=3 gives A006095.
%Y Cf. A058878, A281123, A340312, A340263.
%K nonn,tabf
%O 0,11
%A _Andrew Howroyd_, Jan 09 2021
|
