DigitSums

static class SumFinder
{
    static Dictionary<(int, int), long> record = new Dictionary<(int, int), long>();

    public static long CountSums(int n)
    {
        long ret = 0;
        for (int iLength = 1; iLength <= n; iLength++)
        {
            ret += CountSums(iLength, n);
        }
        return ret;
    }

    public static long CountSums(int length, int n)
    {
        if (n <= 0 || length < (n + 8) / 9)
        {
            return 0;
        }
        if (length == 1)
        {
            return 0 < n && n < 10 ? 1 : 0;
        }
        if (record.ContainsKey((length, n)))
        {
            return record[(length, n)];
        }

        long total = 0;

        for (int iLastDigit = 1; iLastDigit < 10; iLastDigit++)
        {
            total += CountSums(length - 1, n - iLastDigit);
        }
        record[(length, n)] = total;
        return total;
    }
}