# 點算的奧秘：二元母函數

Σ0 ≤ r < ∞ arxr    (1)

(1 + y + y2 + ...) × (1 + y + y2 + ...) × (1 + y + y2 + ...)

[1 + xy + (xy)2 + ...] × [1 + x2y + (x2y)2 + ...] × [1 + x3y + (x3y)2 + ...]    (2)

[1 + xmy + (xmy)2 + ...] × [1 + xm+1y + (xm+1y)2 + ...] × ... [1 + xMy + (xMy)2 + ...]    (3)

[1 + Nmxmy + (Nmxmy)2 + ...] × [1 + Nm+1xm+1y + (Nm+1xm+1y)2 + ...] × ... [1 + NMxMy + (NMxMy)2 + ...]    (4)

[1 + N1xy + (N1xy)2 + ...] × [1 + N2x2y + (N2x2y)2 + ...] × [1 + N3x3y + (N3x3y)2 + ...]

N24 + N1N22N3 + N12N32

[1 + N1xy + (N1xy)2 + ...] × [1 + N2x2y + (N2x2y)2 + ...] × [1 + N3x3y + (N3x3y)2 + ...] ×
[1 + N4x4y + (N4x4y)2 + ...] × [1 + N5x5y + ...] × [1 + N6x6y + ...] × ...

N2N32 + N22N4 + N1N3N4 + N1N2N5 + N12N6

[N2x2y + (N2x2y)2 + (N2x2y)3] × [1 + N4x4y + (N4x4y)2 + ...] × [1 + N6x6y + ...] × [1 + N8x8y + ...] × ...

N2N8y2 + N2N42y3 + N22N6y3 + N23N4y4

[1 + N2x2y + (N2x2y)2 + ...] × [1 + N4x4y + (N4x4y)2 + ...] × [1 + N6x6y + (N6x6y)2 + ...]

N23x6 + N22N4x8 + (N2N42 + N22N6)x10 + (N43 + N2N4N6)x12 + (N42N6 + N2N62)x14 + N4N62x16 + N63x18

(1 + x + x2 + ... )k

Σ1 ≤ k < ∞ [y(1 + x + x2 + ... )]k = Σ1 ≤ k < ∞ [y / (1 − x)]k

C(3 + 5 − 1, 5) = 21

C(3 + 5 − 1, 5)正是以前介紹過的從3類物件中可重覆地抽5個出來的抽法總數的計算公式。□

Σ1 ≤ k < ∞ [y(1 + x + x2/2! + ... )]k = Σ1 ≤ k < ∞ (yex)k