A104471 Triangle of degree numbers for certain polynomials.

1, 2, 1, 3, 4, 1, 4, 5, 4, 1, 5, 8, 8, 4, 1, 6, 9, 11, 8, 4, 1, 7, 12, 12, 15, 8, 4, 1, 8, 13, 18, 16, 15, 8, 4, 1, 9, 16, 19, 22, 21, 15, 8, 4, 1, 10, 17, 22, 23, 27, 21, 15, 8, 4, 1, 11, 20, 26, 30, 28, 33, 21, 15, 8, 4, 1, 12, 21, 29, 34, 35, 34, 33, 21, 15, 8, 4, 1, 13, 24, 30, 37, 39, 41

Offset: 1

Author

Wolfdieter Lang, Mar 24 2005

Keywords

• nonn
• easy
• tabl

Comments

This is the rectangular array ay(N,m):=sum(j*floor(N/j),j=1..m), m>=1, N>=1, written as a triangle a(n,m):=ay(n-m+1,m).

If A(j,x) is a polynomial of exact degree N(j):=floor(N/j), with some N>=1 and j=1,2,..., then F(m,x):=product(A(j,x^j),j=1..m) has degree A(N,m). This product appears in Fine's identity (on the lhs) if a finite product with m factors is taken. See the Riordan reference p. 182 eq.(20).

This choice of the degree N(j) guarantees that in F(m,x) all coefficients of x^n for n=0,...,N are correctly given. Due to Fine's identity (on the rhs) the coefficient of x^n of F(m,x) is given by the sum over all partitions of n with number of parts m of the product(a(j,k(j)),j=1..m), where a(j,p) is [x^p]A(j,x) and k(j)>=0 is the exponent of j in the considered partition of n into m parts. If n

Examples

			[1];[2,1];[3,4,1];[4,5,4,1];[5,8,8,4,1];...
a(5,3)=ay(3,3)=8 because for N=3 and m=3 one has N(1)=3, N(2)=1 and N(3)=1 and (a(1,0)+ a(1,1)*x^1+ ...+a(1,3)*x^3)*(a(2,0)+a(2,1)*x^2)*(a(3,0)+a(3,1)*x^3) has exact degree 3+2+3=8.
		

References

• J. Riordan, Combinatorial Identities, Wiley, 1968.

Links

• W. Lang: First 10 rows and array ay(N,m).

Crossrefs

See triangle A104472 where repeated numbers are omitted. a(2*n-1, n)=A024916(n).

Formula

a(n, m)= sum(j*floor((n-m+1)/j), j=1..m), n>=m>=1, else 0.

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