On Sylvester’s denumerants

Abstract The paper deals with the problem of calculating Sylvester’s denumerant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>b</m:mi> <m:mo>;</m:mo> <m:msub> <m:mi>a</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>a</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>a</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {D(b;a_{1},a_{2},\dots,a_{n})} , i.e., the number <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>b</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {D(b)} of integer non-negative solutions of an equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msubsup> <m:mo largeop="true" symmetric="true">∑</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:msubsup> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mi>b</m:mi> </m:mrow> </m:math> {\sum_{i=1}^{n}a_{i}x_{i}=b} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {n\geq 2} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>a</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>a</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:math> {a_{1},a_{2},\dots,a_{n}} are natural numbers and b is a non-negative integer. Unlike traditional methods of finding denumerants, in our study we employ only elementary methods, mainly the properties of the Kronecker function and the elements of combinatorics. The formula is derived to express <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>b</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {D(b)} , for an integer non-negative b , coprime <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>a</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>a</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:math> {a_{1},a_{2},\dots,a_{n}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>s</m:mi> <m:mo>≠</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {s\neq 0}