Generate Pascal's Pyramid

Question

Pascal's Pyramid is an extension of Pascal's Triangle to the third dimension. Starting with a 1 as the apex, the elements of each successive layer can be determined by summing the three numbers that are above it. The nth layer contains the coefficients of (a + b + c)^(n-1).

You are to create a program that, given an input n, outputs the first n layers of Pascal's Pyramid. There must be two newlines between the different layers, and they must be in order. Within each layer, each row must be on a separate line, with the elements separated by spaces. Apart from that, you may format the pyramid however you like.

Your code should be reasonably fast generating a pyramid with 10 layers.

For example, your program, when given 4 as input, should output something like this:

1

1
1 1

1
2 2
1 2 1

1
3 3
3 6 3
1 3 3 1

May the shortest code win!

Answer 1

Python, 110

l=1
exec"print;n=1;r=0;exec\"c=0;exec'print n,;n*=r-c;c+=1;n/=c;'*r;print n;r+=1;n=n*l/r-n;\"*l;l+=1;"*input()

Original, ungolfed solution:

for layer in range(1, input() + 1):
    print
    n = 1
    for row in range(1, layer + 1):
        for col in range(1, row):
            print n,
            n *= row - col
            n /= col
        print n
        n *= layer - row
        n /= row

Answer 2

Mathematica (112)

For[j = 0, j < 11, j++, For[i = 0, i <= j, i++, Print[Row[Array[Binomial[i, # - 1] &, i + 1], " "]]]; Print[""]]

Older method(166):

b = "stdout"; For[k = 0, k < 11, k++, For[i = 0, i <= k, i++, For[j = 0, j <= i, j++, WriteString[b, Binomial[i, j], " "]]; WriteString[b, "\n"]]; WriteString[b, "\n"]]

The following prints it in a different format, but is only 94 characters:

For[i = 0, i < 11, i++, Print[Column[Table[Binomial[n, k], {n, 0, i}, {k, 0, n}]]]; Print[""]]

Answer 3

Mathematica 65 64

c=Column;c[c@Table[n~Binomial~k,{n,0,p},{k,0,n}]~Table~{p,0,#}]&

Usage

c=Column;c[c@Table[n~Binomial~k,{n,0,p},{k,0,n}]~Table~{p,0,#}]&@5

Answer 4

Python (134 128)

Factorials are probably more space-efficient, but I wanted to be fancy.

for n in range(1,input()+1):
 a=r=1.
 while a:
  b=a;j=1.;a*=(n-r)/r
  while b:print int(b),;b*=(r-j)/j;j+=1
  print;r+=1
 print

Uses the ratio property of Pascal's Triangle.
e.x. 1 4 6 4 1, 1:4, 4:6, 6:4, 4:1 = 1:4, 2:3, 3:2, 4:1

for n in range(1,input()+1): #Generate layers
 a=r=1.               #a is the first value of a row in a layer; r represents the row
 while a:          #when a is 0, there are no more rows in the pyramid.
  b=a;i=1.;a*=(n-r)/r #b will calculate the values of that row
                      #use ratio property to determine next value of a
                      #i is a counter representing the ith item in row r
  while b:print int(b),;b*=(r-i)/i;i+=1 #Output values using ratio property
  print;r+=1          #increment row; loop
 print

Answer 5

k 31

q)k)f:{,/{x**|x}{x{0+':x,0}\1}@'!x}
q)f 4
1
1
1 1
1
2 2
1 2 1
1
3 3
3 6 3
1 3 3 1

Answer 6

q)k)f:{,/{p**|p:x{0+':x,0}\1}'!x}

q)f 4

1  
1  
1 1  
1  
2 2  
1 2 1  
1  
3 3  
3 6 3  
1 3 3 1