A340030 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.

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, 105, 35, 0, 0, 1, 1, 0, 0, 155, 1085, 5208, 22568, 82615, 247845, 628680, 1383096, 2648919, 4414865, 6440560, 8280720, 9398115, 9398115, 8280720, 6440560, 4414865, 2648919, 1383096, 628680, 247845, 82615, 22568, 5208, 1085, 155, 0, 0, 1

Offset

0,11

Comments

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.

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.

Also the coefficients of polynomials p_{n}(x) which have the representation

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

Links

Andrew Howroyd, Table of n, a(n) for n = 0..2046 (rows 0..10)

Wikipedia, Hypergraph.

Formula

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.

T(n,2*k) + T(n,2*k+1) = binomial(2^n-1, k)/2^n = A281123(n,k).

T(n, k) = T(n, 2^n-1-k) for n >= 2.

Example

Triangle begins:
[0]  1;
[1]  1, 0;
[2]  1, 0, 0,  1;
[3]  1, 0, 0,  7,   7,   0,   0,   1;
[4]  1, 0, 0, 35, 105, 168, 280, 435, 435, 280, 168, 105, 35, 0, 0, 1;

Prog

(PARI)
T(n, k) = {(binomial(2^n-1, k) + (-1)^((k+1)\2)*(2^n-1)*binomial(2^(n-1)-1, k\2))/2^n}
{ for(n=0, 5, print(vector(2^n, k, T(n, k-1)))) }

Crossrefs

Rows 3..8 are A002394, A010085, A010086, A010087, A010088, A010089.

Row sums are A016031(n+1).

Column k=3 gives A006095.

Cf. A058878, A281123, A340312, A340263.

Sequence in context: A284210 A371078 A002394 * A274210 A105167 A217227

Adjacent sequences: A340027 A340028 A340029 * A340031 A340032 A340033

Keyword

nonn,tabf

Author

Andrew Howroyd, Jan 09 2021

Status

approved