A377065 Number of 3 X n 0..2 matrices with row sums n and column sums 3 up to permutations of rows.

1, 1, 2, 6, 30, 166, 981, 5937, 36646, 229350, 1451757, 9274057, 59699729, 386798777, 2520034050, 16497343046, 108454221206, 715629888822, 4737625385061, 31456633327905, 209418369288865, 1397521222483385, 9346484009527370, 62632803958053870, 420481623373564025

Offset: 0

Author

Andrew Howroyd, Oct 15 2024

Keywords

• nonn

Comments

Also, the number of 3 X n {-1,0,1} matrices with all rows and columns summing to zero up to permutations of rows.

Examples

			The a(2) = 2 matrices are:
  [1 1]  [2 0]
  [1 1]  [0 2]
  [1 1]  [1 1]
The a(3) = 6 matrices are:
  [1 1 1]  [2 1 0]  [2 0 1]  [1 2 0]  [2 1 0]  [2 0 1]
  [1 1 1]  [0 1 2]  [0 2 0]  [1 0 2]  [1 0 2]  [1 2 0]
  [1 1 1]  [1 1 1]  [1 1 1]  [1 1 1]  [0 2 1]  [0 1 2]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500

Crossrefs

Row n=3 of A377063.

Cf. A172634, A377066, A377067.

Programs

PARI

a(n)={(5+sum(i=0, n, sum(j=0, i, (-1)^(n-i)*binomial(n, i)*binomial(i, j)^3)))/6}

Formula

a(n) = (A172634(n) - 1)/6 + 1.

a(n) = (5 + Sum_{i=0..n} Sum_{j=0..i} (-1)^(n-i)*binomial(n, i)*binomial(i, j)^3)/6.