Python 杨辉三角形的若干种求法

收集了杨辉三角形的若干种求法，所有方法用以下函数输出测试结果：

>>> def test(func, n, out=True):from time import timestart = time()if not out:t=(func(n))else:for i in range(1,n+1):print(f'Y({i:>2}) = {func(i)}')print(time()-start)

方法一：

>>> def Yh1(n):t=L=[1]while n-1:n-=1t=L+[t[i]+t[i+1] for i in range(len(t)-1)]+Lreturn t>>> test(Yh1,17)
Y( 1) = [1]
Y( 2) = [1, 1]
Y( 3) = [1, 2, 1]
Y( 4) = [1, 3, 3, 1]
Y( 5) = [1, 4, 6, 4, 1]
Y( 6) = [1, 5, 10, 10, 5, 1]
Y( 7) = [1, 6, 15, 20, 15, 6, 1]
Y( 8) = [1, 7, 21, 35, 35, 21, 7, 1]
Y( 9) = [1, 8, 28, 56, 70, 56, 28, 8, 1]
Y(10) = [1, 9, 36, 84, 126, 126, 84, 36, 9, 1]
Y(11) = [1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1]
Y(12) = [1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1]
Y(13) = [1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1]
Y(14) = [1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1]
Y(15) = [1, 14, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, 1]
Y(16) = [1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1]
Y(17) = [1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1]
0.2859833240509033
>>> 
>>> test(Yh1,323,False)
0.015624284744262695
>>> test(Yh1,1000,False)
0.2491614818572998
>>> test(Yh1,2000,False)
1.452894926071167
>>> test(Yh1,3000,False)
4.124391317367554
>>> 

方法二：

>>> def Yh2(n):t=[[1],[1,1]]i=0while i+2>> test(Yh2,17)
Y( 1) = [1]
Y( 2) = [1, 1]
Y( 3) = [1, 2, 1]
Y( 4) = [1, 3, 3, 1]
Y( 5) = [1, 4, 6, 4, 1]
Y( 6) = [1, 5, 10, 10, 5, 1]
Y( 7) = [1, 6, 15, 20, 15, 6, 1]
Y( 8) = [1, 7, 21, 35, 35, 21, 7, 1]
Y( 9) = [1, 8, 28, 56, 70, 56, 28, 8, 1]
Y(10) = [1, 9, 36, 84, 126, 126, 84, 36, 9, 1]
Y(11) = [1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1]
Y(12) = [1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1]
Y(13) = [1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1]
Y(14) = [1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1]
Y(15) = [1, 14, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, 1]
Y(16) = [1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1]
Y(17) = [1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1]
0.29276371002197266
>>> 
>>> test(Yh2,1000,False)
0.3641676902770996
>>> test(Yh2,2000,False)
1.9820797443389893
>>> test(Yh2,3000,False)
5.548851013183594
>>> 

方法三：

>>> def Yh3(n):t=[_*[1] for _ in range(1,n+1)]for i in range(2,n):for j in range(1,i):t[i][j]=t[i-1][j-1]+t[i-1][j]return t[n-1]>>> test(Yh3,17)
Y( 1) = [1]
Y( 2) = [1, 1]
Y( 3) = [1, 2, 1]
Y( 4) = [1, 3, 3, 1]
Y( 5) = [1, 4, 6, 4, 1]
Y( 6) = [1, 5, 10, 10, 5, 1]
Y( 7) = [1, 6, 15, 20, 15, 6, 1]
Y( 8) = [1, 7, 21, 35, 35, 21, 7, 1]
Y( 9) = [1, 8, 28, 56, 70, 56, 28, 8, 1]
Y(10) = [1, 9, 36, 84, 126, 126, 84, 36, 9, 1]
Y(11) = [1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1]
Y(12) = [1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1]
Y(13) = [1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1]
Y(14) = [1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1]
Y(15) = [1, 14, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, 1]
Y(16) = [1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1]
Y(17) = [1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1]
0.2826821804046631
>>> 
>>> test(Yh3,1000,False)
0.4663882255554199
>>> test(Yh3,2000,False)
2.530691385269165
>>> test(Yh3,3000,False)
6.483311176300049
>>> 

方法四：

>>> def Yh4(n):L=[1]while n-1:n -= 1L.append(0)L = [L[i-1] + L[i] for i in range(len(L))]return L>>> test(Yh4,17)
Y( 1) = [1]
Y( 2) = [1, 1]
Y( 3) = [1, 2, 1]
Y( 4) = [1, 3, 3, 1]
Y( 5) = [1, 4, 6, 4, 1]
Y( 6) = [1, 5, 10, 10, 5, 1]
Y( 7) = [1, 6, 15, 20, 15, 6, 1]
Y( 8) = [1, 7, 21, 35, 35, 21, 7, 1]
Y( 9) = [1, 8, 28, 56, 70, 56, 28, 8, 1]
Y(10) = [1, 9, 36, 84, 126, 126, 84, 36, 9, 1]
Y(11) = [1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1]
Y(12) = [1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1]
Y(13) = [1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1]
Y(14) = [1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1]
Y(15) = [1, 14, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, 1]
Y(16) = [1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1]
Y(17) = [1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1]
0.29657435417175293
>>>
>>> test(Yh4,1000,False)
0.24535727500915527
>>> test(Yh4,2000,False)
1.377683162689209
>>> test(Yh4,3000,False)
3.977062702178955
>>> 

方法五：

>>> def Yh5(n):L = [1]for i in range(1,n):L+=[0]L=[L[j]+k for j,k in enumerate(L[::-1])]return L>>> test(Yh5,17)
Y( 1) = [1]
Y( 2) = [1, 1]
Y( 3) = [1, 2, 1]
Y( 4) = [1, 3, 3, 1]
Y( 5) = [1, 4, 6, 4, 1]
Y( 6) = [1, 5, 10, 10, 5, 1]
Y( 7) = [1, 6, 15, 20, 15, 6, 1]
Y( 8) = [1, 7, 21, 35, 35, 21, 7, 1]
Y( 9) = [1, 8, 28, 56, 70, 56, 28, 8, 1]
Y(10) = [1, 9, 36, 84, 126, 126, 84, 36, 9, 1]
Y(11) = [1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1]
Y(12) = [1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1]
Y(13) = [1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1]
Y(14) = [1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1]
Y(15) = [1, 14, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, 1]
Y(16) = [1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1]
Y(17) = [1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1]
0.2609119415283203
>>>
>>> test(Yh5,1000,False)
0.2201399803161621
>>> test(Yh5,2000,False)
1.249098300933838
>>> test(Yh5,3000,False)
3.7358851432800293
>>> 

方法六：

>>> def Yh6(n):L=[1]for _ in range(n-1):L=[sum(_) for _ in zip([0]+L,L+[0])]return L>>> test(Yh6,17)
Y( 1) = [1]
Y( 2) = [1, 1]
Y( 3) = [1, 2, 1]
Y( 4) = [1, 3, 3, 1]
Y( 5) = [1, 4, 6, 4, 1]
Y( 6) = [1, 5, 10, 10, 5, 1]
Y( 7) = [1, 6, 15, 20, 15, 6, 1]
Y( 8) = [1, 7, 21, 35, 35, 21, 7, 1]
Y( 9) = [1, 8, 28, 56, 70, 56, 28, 8, 1]
Y(10) = [1, 9, 36, 84, 126, 126, 84, 36, 9, 1]
Y(11) = [1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1]
Y(12) = [1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1]
Y(13) = [1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1]
Y(14) = [1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1]
Y(15) = [1, 14, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, 1]
Y(16) = [1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1]
Y(17) = [1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1]
0.173600435256958
>>> test(Yh6,1000,False)
0.14240050315856934
>>> test(Yh6,2000,False)
0.5334012508392334
>>> test(Yh6,3000,False)
1.3280022144317627
>>> 

递归法：

>>> def Yh(n):if n==1: return [1]t = Yh(n-1)+[0]return [sum(z) for z in zip(t,t[::-1])]'''
或者：
>>> def Yh(n):if n==1: return [1]t = Yh(n-1)return [1]+[t[i]+t[i+1] for i in range(len(t)-1)]+[1]
'''>>> test(Yh,17)
Y( 1) = [1]
Y( 2) = [1, 1]
Y( 3) = [1, 2, 1]
Y( 4) = [1, 3, 3, 1]
Y( 5) = [1, 4, 6, 4, 1]
Y( 6) = [1, 5, 10, 10, 5, 1]
Y( 7) = [1, 6, 15, 20, 15, 6, 1]
Y( 8) = [1, 7, 21, 35, 35, 21, 7, 1]
Y( 9) = [1, 8, 28, 56, 70, 56, 28, 8, 1]
Y(10) = [1, 9, 36, 84, 126, 126, 84, 36, 9, 1]
Y(11) = [1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1]
Y(12) = [1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1]
Y(13) = [1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1]
Y(14) = [1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1]
Y(15) = [1, 14, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, 1]
Y(16) = [1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1]
Y(17) = [1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1]
0.3664591312408447
>>> test(Yh,1000,False)
0.4101085662841797# test(Yh,1024,False) 超最大递归深度

组合公式法：

>>> def Combin(n,i):m,t=min(i,n-i),1for j in range(0,m):t*=(n-j)/(m-j)return t>>> def Yh(n):return [round(Combin(n-1,i)) if n<1000 else Combin(n-1,i) for i in range(n)]>>> test(Yh,17)
Y( 1) = [1]
Y( 2) = [1, 1]
Y( 3) = [1, 2, 1]
Y( 4) = [1, 3, 3, 1]
Y( 5) = [1, 4, 6, 4, 1]
Y( 6) = [1, 5, 10, 10, 5, 1]
Y( 7) = [1, 6, 15, 20, 15, 6, 1]
Y( 8) = [1, 7, 21, 35, 35, 21, 7, 1]
Y( 9) = [1, 8, 28, 56, 70, 56, 28, 8, 1]
Y(10) = [1, 9, 36, 84, 126, 126, 84, 36, 9, 1]
Y(11) = [1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1]
Y(12) = [1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1]
Y(13) = [1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1]
Y(14) = [1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1]
Y(15) = [1, 14, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, 1]
Y(16) = [1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1]
Y(17) = [1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1]
0.26958274841308594
>>> 
>>> test(Yh,1000,False)
0.05560708045959473
>>> test(Yh,2000,False)
0.2970747947692871
>>> test(Yh,3000,False)
0.7152261734008789
>>> 

类方法：

class Yh():def __init__(self,n=None):self.data = [1]if n!=None: self.data=Yh()*ndef __repr__(self):return f'{self.data}'def __mul__(self,n):while n>1:n-=1self.data.append(0)self.data=[i[0]+i[1] for i in zip(self.data,self.data[::-1])]return self.data__rmul__ = __mul__def List(n):i=0while i>> test(Yh,17)
Y( 1) = [1]
Y( 2) = [1, 1]
Y( 3) = [1, 2, 1]
Y( 4) = [1, 3, 3, 1]
Y( 5) = [1, 4, 6, 4, 1]
Y( 6) = [1, 5, 10, 10, 5, 1]
Y( 7) = [1, 6, 15, 20, 15, 6, 1]
Y( 8) = [1, 7, 21, 35, 35, 21, 7, 1]
Y( 9) = [1, 8, 28, 56, 70, 56, 28, 8, 1]
Y(10) = [1, 9, 36, 84, 126, 126, 84, 36, 9, 1]
Y(11) = [1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1]
Y(12) = [1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1]
Y(13) = [1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1]
Y(14) = [1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1]
Y(15) = [1, 14, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, 1]
Y(16) = [1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1]
Y(17) = [1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1]
0.19911837577819824
>>>
>>> test(Yh,1000,False)
0.2420802116394043
>>> test(Yh,2000,False)
1.3822360038757324
>>> test(Yh,3000,False)
4.064407110214233
>>>
>>>  # 副产品：除了像函数一样使用外，还能用乘法表示项数，也可以直接列表
>>> Yh(11)
[1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1]
>>> Yh()*12
[1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1]
>>> 12*Yh()
[1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1]
>>> Yh.List(10)
[1]
[1, 1]
[1, 2, 1]
[1, 3, 3, 1]
[1, 4, 6, 4, 1]
[1, 5, 10, 10, 5, 1]
[1, 6, 15, 20, 15, 6, 1]
[1, 7, 21, 35, 35, 21, 7, 1]
[1, 8, 28, 56, 70, 56, 28, 8, 1]
[1, 9, 36, 84, 126, 126, 84, 36, 9, 1]
>>> 

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