Euler Polynomials and Combinatoric Convolution Sums of Divisor Functions with Even Indices

and Applied Analysis 3 Table 1 n 2 3 4 5 6 7 8 9 10 11 12 T(n) 2 10 24 50 96 148 224 342 452 606 832


Application and Perspective
2.1.Branch and Leaves Pattern.In real-time virtual system, realistic and efficient generation and expression of plants composing a broad terrain is a continuing problem [7,8].The procedural modelling method using convolution sums of divisor functions (MCD) was suggested for a variety of natural trees in a virtual ecosystem [9].The basic structure of MCD is that it defines the growth grammar including the branch propagation, a growth pattern of branches and leaves, and a process of growth deformation for various generations of tree.In Figure 1, we illustrate the MCD for the divisor trees.

Color of Leaves.
Leaves are important elements that constitute plants [12].In [13] we see the leaves model of convolution sum of divisor functions.First, we fix the random color variation step  and then we compute convolution sums by use of (6) as shown in Figure 2. We denote by (  ,   ) the start and destination colors; output colors in phase are determined by interpolating convolution sums.Next, based on the minimum and maximum values of the elements of convolution sums of divisor functions, normalized weighted values between 0 and 1 are calculated and then each color table is organized as follows: The noise map that utilizes Perlin noise is used in the creation of cloud textures for soft color changes.Here,  size is the number of values that are not duplicated among the elements of convolution sums calculated in the selected table;  max  ,  min  are the maximum and minimum noise values; and  step indicates the section of noise values influenced by a single element of convolution sums.Once these parameters are fixed, the leaf color is determined by finding the index "" and the corresponding color "   " in the color table comprised of divisor functions that correspond to the noise color in th row and th column [13].Figure 2 represents the image of a color table set created by calculating the color table of each section.Theorem 1 gives the number of values of leaf color and Theorem 3 gives pattern of branch and leaves in the same model.In real-time ecosystem, efficient and fast time is requested.When  is large, the MCD that uses ( 7) is complicate to compute directly in reasonable time.Our results (8), (9), and (10) give simple formulas for efficient time for them.In the forthcoming papers, we will investigate more applications of real-time virtual ecosystem.

Proof of Main Results.
To prove Theorem 1, we establish Lemmas 4, 6, 7, and 9 and Corollary 5 and we recall Proposition 8.
By [14,Theorem 3] and Lemma 4 we get the corollary.
Lemma 6.For any integers  and  ≥ 1, one has Proof.Write the series as follows: This completes the proof of (17).On the other hand, (18) may be rewritten as From ( 17) we obtain This completes the proof of the lemma.
Lemma 7.For any integers  ≥ 1 and  ≥ 2, one has Proof.We observe that From (18), we get We then get lemma.
Abstract and Applied Analysis 5 We recall Proposition 8 to be modified in Lemma 9.
Lemma 9.For any ,  ∈ N one has Proof.First, we check that From Lemma 7 and Proposition 8, our claim follows.
To end the proof of Theorem 1, we observe that