Number Patterns

# Trigonometric Expansion

Use the Maple command expand to generate the following formulae:

The coefficients can be arranged in a table

 1 1 -1 0 2 0 0 0 0 0 0 0 0 -3 0 4 0 0 0 0 0 0 1 0 -8 0 8 0 0 0 0 0 0 5 0 -20 0 16 0 0 0 0 -1 0 18 0 -48 0 32 0 0 0 0 -7 0 56 0 -112 0 64 0 0 1 0 -32 0 160 0 -256 0 128 0 0 9 0 -120 0 432 0 -576 0 256

How is this table generated?

# Expressing in Terms of

Use the Maple command combine to generate the following formulae:

The coefficients can be arranged in a table

 1 1 0 1 0 0 0 0 0 0 0 0 3 0 1 0 0 0 0 0 0 3 0 4 0 1 0 0 0 0 0 0 10 0 5 0 1 0 0 0 0 10 0 15 0 6 0 1 0 0 0 0 35 0 21 0 7 0 1 0 0 35 0 56 0 28 0 8 0 1 0 0 126 0 84 0 36 0 9 0 1

How is this table generated?

# Successive Derivatives of

Define Then

The coefficients can be arranged in a table

 1 1 -1 0 0 0 0 0 0 1 -2 2 0 0 0 0 0 1 -3 6 -6 0 0 0 0 1 -4 12 -24 24 0 0 0 1 -5 20 -60 120 -120 0 0 1 -6 30 -120 360 -720 720 0 1 -7 42 -210 840 -2520 5040 -5040

How is this table generated?

# Same Formula, Different Pattern

It is clear that in general we have

for some polynomial function of degree Indeed from

we see that

for Thus we have the recurrence relations

for

The following table is generated in view of these relations:

 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 -2 3 0 1 0 0 0 9 -8 6 0 1 0 0 -44 45 -20 10 0 1 0 265 -264 135 -40 15 0 1 -1854 1855 -924 315 -70 21 0

# Factors of

Applying the Factor command, we have

Can you see the general pattern?