A100344 Gives the i-th coefficient M(k,i) of the decomposition of the polynomials B(k,X^2) in the basis of all B(i,X), where B(i,X) is the i-th binomial polynomial: B(i,X) = X(X-1)...(X-i+1)/i! for any i > 0 and B(0,X) = 1 by definition.

1, 0, 1, 2, 0, 0, 6, 18, 12, 0, 0, 4, 72, 248, 300, 120, 0, 0, 1, 123, 1322, 4800, 7800, 5880, 1680, 0, 0, 0, 126, 3864, 32550, 121212, 235200, 248640, 136080, 30240

Offset: 0

Author

Jean Francis Michon, Nov 18 2004

Keywords

• nonn
• tabl

Comments

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).

Examples

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

Links

• J. F. Michon, J.-B. Yunes and P. Valarcher, On maximal QROBDD's of Boolean functions, Theor. Inform. Appl. 39 (2005), no. 4, 677-686.

Crossrefs

Cf. for binomial polynomials: A080959.

Formula

M(0, 0) = 1 and, for all i > 0, M(0, i) = 0. Let M(k, i) = 0 if all i < 0 and all k for ease. Then, for all k > 0, i > 0: M(k, i)= [(i^2-k+1)M(k-1, i) + i(2i-1)M(k-1, i-1) + i(i-1)M(k-1, i-2) ]/k.