Denumerant

The number $D(n; a_1,\ldots,a_k)$ of partitions of an integer $n$ into parts equal to $a_1,\ldots,a_m$, i.e. the number of solutions in non-negative integers $x_1,\ldots,x_m$ of the equation $$ n = a_1 x_1 + \cdots + a_m x_m \ . $$ The generating function of the denumerants is $$ D(z; a_1,\ldots,a_m) = \sum_n D(n;a_1,\ldots,a_m) z^n = \frac{1}{\left({1-z^{a_1}}\right)\cdots\left({1-z^{a_m}}\right)} \ . $$

The simplest method of computing a denumerant is by Euler's recurrence relation: $$ D(n;1,\ldots,k) - D(n-k;1,\ldots,k) = D(n;1,\ldots,k-1) \ . $$

Explicit formulas for certain denumerants may be obtained from the following theorem: If $A$ is the least common multiple of the numbers $a_1,\ldots,a_m$, then the denumerant $$ D(An+b;a_1,\ldots,a_m) \ ;\ \ b=0,\ldots,A-1 $$ is a polynomial of degree $(m-1)$ with respect to $n$.

References

[1]	J. Riordan, "An introduction to combinational analysis" , Wiley (1958)