**
Number Patterns**

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?

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?

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?

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 |

Applying the **Factor** command, we have

Can you see the general pattern?