cp's OEIS Frontend

Except for the first term, positive values of x (or y) satisfying x^2 - 20xy + y^2 + 396 = 0. - Colin Barker, Feb 28 2014

The binomial polynomials are a basis of the space of all polynomials and the decomposition of a polynomial in this basis is called its Mahler's expansion. So the sequence gives the Mahler's expansion of the binomial polynomials composed with "squaring".

For example:

B(0,X^2) = 1*B(0,X)

B(1,X^2) = 0*B(0,X)+1*B(1,X)+2*B(2,X)

B(2,X^2) = 0*B(0,X)+0*B(1,X)+6*B(2,X)+18*B(3,X)+12*B(4,X)

The coefficients may be written in a "Pascal's triangle" arrangement:

1

0 1 2

0 0 6 18 12

0 0 4 72 248 300 120

0 0 1 1 123 1322 4800 7800 5880 1680

They are always < binomial(i^2, k) or equal to it when i^2+1 > k > (i-1)^2. They are 0 if i > 2k or k > i^2.

They have a combinatorial interpretation if i > 0. Let the set I={1,...,i} and I X I the set of pairs, M(k,i) is the number of subsets with k pairs in I X I such that any element of I appears as a coordinate in at least one pair.

Example: M(2,2) = 6 because all subsets with 2 elements in IxI = {(1,1),(1,2),(2,1),(2,2)} satisfy the property and there are 6 such subsets.

The M(k,i) sequence allows the enumeration of quasi-reduced ordered binary decision diagram (QROBDD) canonically associated to Boolean functions (see references).