Pascal's Alternating Triangle

Question

Pascal's triangle is generated by starting with 1 and having each row formed from successive additions. Here, instead, we're going to form a triangle by alternating multiplication and addition.

We start row 1 with just a solitary 1. Thereafter, addition is done on the odd rows, and multiplication is done on the even rows (1-indexed). When performing the addition step, assume the spaces outside of the triangle are filled with 0s. When performing the multiplication step, assume that the outside is filled with 1s.

Here's the full triangle down to 7 rows. The * or + on the left shows what step was performed to generate that row.

1                1
2 *            1   1
3 +          1   2   1
4 *        1   2   2   1
5 +      1   3   4   3   1
6 *    1   3  12  12   3   1
7 +  1   4  15  24  15   4   1

Challenge

Given input n, output the nth row of this triangle.

Rules

• You may choose to 0-index instead, but then please realize that the addition and multiplication rows must flip-flop, so that the exact same triangle is generated as above. Please state in your submission if you choose to do this.
• The input and output can be assumed to fit in your language's native integer type.
• The input and output can be given in any convenient format.
• Either a full program or a function are acceptable. If a function, you can return the output rather than printing it.
• If possible, please include a link to an online testing environment so other people can try out your code!
• Standard loopholes are forbidden.
• This is code-golf so all usual golfing rules apply, and the shortest code (in bytes) wins.

Examples

Showing two possible examples of output out of many: a list, or a space separated string.

4
[1, 2, 2, 1]

8
"1 4 60 360 360 60 4 1"

Answer 1

Pascal, 249 247 233 bytes

Well, this is Pascal's alternating triangle.

_{1 byte saved thanks to @Mr.Xcoder}

function f(n,k:integer):integer;begin if((k<1)or(k>n)or(n=1))then f:=n mod 2 else if n mod 2=0then f:=f(n-1,k-1)*f(n-1,k)else f:=f(n-1,k-1)+f(n-1,k)end;
procedure g(n:integer);var k:integer;begin for k:=1to n do write(f(n,k),' ')end;

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Answer 2

Python 2, 97 93 86 81 78 bytes

-4 bytes thanks to Rod. -10 bytes thanks to Halvard Hummel.

f=lambda n:n and[[i+j,i*j][n%2]for i,j in zip([n%2]+f(n-1),f(n-1)+[n%2])]or[1]

0-indexed.

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Answer 3

Python 2, 96 89 87 bytes

• 2 bytes Thanks to Mr. Xcoder: s=[1]

• A bit different than totallyhuman's answer

s=a=[1]
for i in range(1,input()):a=s+[[k+l,k*l][i%2]for k,l in zip(a[1:],a)]+s
print a

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Answer 4

Jelly, 17 12 bytes

µ×+LḂ$?Ḋ1;µ¡

This is a full program (or niladic link) that takes input from STDIN.

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How it works

µ×+LḂ$?Ḋ1;µ¡  Main link. No arguments. Implicit argument: 0

          µ¡  Start a monadic chain and apply the ntimes quick to the previous one.
              This reads an integer n from STDIN and executes the previous chain n
              times, with initial argument 0, returning the last result.
µ             Start a monadic chain. Argument: A (array or 0)
       Ḋ          Dequeue; yield A without its first element.
   LḂ$?           If the length of A is odd:
 ×                    Multiply A and dequeued A.
                  Else:
  +                   Add A and dequeued A.
        1;        Prepend a 1 to the result.

Answer 5

CJam, 25 bytes

{1a\{2%!_2$+\{.*}{.+}?}/}

0-indexed.

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Explanation

This is an anonymous block that takes the number from the stack and leaves the result on the stack.

1a                        Push [1].
  \                       Bring the number to the top.
   {                 }/   For reach number 0 .. arg-1, do:
    2%!                    Push 0 if even, 1 if odd.
       _                   Copy that.
        2$+                Copy the list so far and prepend the 0 or 1 to it.
           \               Bring the 0 or 1 back to the top.
            {.*}{.+}?      If 1, element-wise multiplication. If 0, element-wise addition.

Answer 6

Haskell, 76 72 bytes

0-indexed solution:

(p!!)
p=[1]:[zipWith o(e:l)l++[1]|(l,(o,e))<-zip p$cycle[((*),1),((+),0)]]

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Explanation

p recursively defines the alternating triangle, the base case/first element of it is [1]

p=[1]:[                                                            ]

It then builds the triangle by taking the previous line (l). To know what to do with it we need to keep track of the correct operator (o) and the corresponding neutral element (e):

                           |(l,(o,e))<-zip p$cycle[((*),1),((+),0)]

From this build the new line by duplicating the line and for one copy we prepend the neutral element, zip them with the operator and append a 1:

       zipWith o(e:l)l++[1]

Answer 7

Mathematica, 92 bytes

(s={i=1};While[i<#,s=Flatten@{1,{Tr/@#,Times@@@#}[[i~Mod~2+1]]&@Partition[s,2,1],1};i++];s)&

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Answer 8

R, 108 98 bytes

-10 bytes by replacing the actual multiply sign with a plus sign. Please forgive me.

f=function(n){if(n<3)return(rep(1,n))else{v=f(n-1)};if(n%%2)`*`=`+`;return(c(1,v[3:n-2]*v[-1],1))}

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Quite pleased with the general method (first time I've aliased a primitive) but I'm sure there's golfing to be done on it yet, especially with the awkward handling of cases where n<3 which leads to a lot of boilerplate.

Answer 9

Husk, 17 16 bytes

!G₅;1¢e*+
:1Sż⁰t

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A 1-indexed solution.

Explanation

The first line is the main function, which calls the helper function on the second line. The helper function is usually called with ₁, but in this case I'm using the overflowing labels feature of Husk: if you refer to a line N in a program with M < N lines, you get line N mod M with modifier function M / N applied to it. The second modifier function is flip, so I'm using ₅ to flip the arguments of the helper function with no additional byte cost.

Here is the helper function.

:1Sż⁰t  Takes a function f and a list x.
   ż    Zip preserving elements of longer list
    ⁰   using function f
  S  t  x and its tail,
:1      then prepend 1.

Here is the main function.

!G₅;1¢e*+  Takes a number n.
      e*+  2-element list of the functions * and +
     ¢     repeated infinitely.
 G         Left scan this list
  ₅        using the flipped helper function
   ;1      with initial value [1].
!          Get n'th element.

Answer 10

C# (.NET Core), 143 134 128 bytes

-4 bytes thanks to Phaeze
-5 bytes thanks to Zac Faragher
-6 bytes thanks to Kevin Cruijssen

n=>{int[]b={1},c;for(int i=0,j;++i<n;b=c)for(c=new int[i+1],c[0]=c[i]=1,j=0;++j<i;)c[j]=i%2<1?b[j-1]+b[j]:b[j-1]*b[j];return b;}

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Explanation:

n =>
{
    int[] b = { 1 }, c;               // Create first layer
    for(int i = 0, j; ++i < n; b = c) // Iterate for every layer, replace last layer with a new one
        for(c = new int[i+1],         // Create new layer
            c[0] = c[i] = 1,          // First and last elements are always 1
            j = 0;
            ++j < i; )                // Replace every element (besides 1st and last)...
                c[j] = i % 2 == 0 ?
                    b[j - 1] + b[j] : // ... with addition...
                    b[j - 1] * b[j];  // ... or multiplication of two from previous layers
    return b;                         // Return latest layer
};

Answer 11

Python 2, 83 bytes

Give some love to exec
0-indexed

l=[1];exec"l[1:]=[[a+b,a*b][len(l)%2]for a,b in zip(l,l[1:])]+[1];"*input()
print l

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Answer 12

Pyth, 22 bytes

Saved tons of byte thanks to @FryAmTheEggman! The initial solution is below.

u++1@,+VGtG*VGtGlG1Q[1

Full Test Suite (0-indexed).

Pyth, 40 38 36 35 bytes

This feels waaaaaaaay too reasonably long. Suggestions are welcome.

K]1VStQ=K++]1m@,sd*hded%N2C,KtK]1;K

Test Suite or Try it online!

Answer 13

J, 32 bytes

[:(1,]{::(}.@,-.)(*;+)[)~/2|]-i.

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Answer 14

Perl 5, 111 + 2 (-na) = 113 bytes

sub t{($r,$c)=@_;!--$r||!$c||$c>=$r?1:eval"t($r,$c)".($r%2?"*":"+")."t($r,$c-1)"}say map{t($F[0],$_).$"}0..$_-1

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Answer 15

Mathematica, 70 bytes

Fold[#2@@@Partition[#,2,1,{-1,1},{}]&,{1},PadRight[{},#,{1##&,Plus}]]&

Try it at the Wolfram sandbox! It doesn't work in Mathics, unfortunately. It's 0-indexed.

Explanation: Partition[#,2,1,{-1,1},{}] takes a list and returns all the two-element sublists, plus 1-element lists for the start and end — e.g., {1,2,3,4} becomes {{1}, {1,2}, {2,3}, {3,4}, {4}}. PadRight[{},#,{1##&,Plus}] makes an alternating list of 1##& (effectively Times) and Plus, whose length is the input number. Then Fold repeatedly applies the partition function with the Pluses and Timeses applied to it, to make the rows of the triangle.

Answer 16

Ruby, 83 82 bytes

->n{a=[1];p=0;n.times{a=[p=1-p,*a,p].each_cons(2).map{|x|x.reduce([:+,:*][p])}};a}

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This is 0-indexed.

Answer 17

Racket, 116 bytes

(define(t n[i 1][r'(1)])(if(= n 1)r(t(- n 1)(- 1 i)(cons 1(append(map(if(> i 0)* +)(cdr r)(reverse(cdr r)))'(1))))))

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Answer 18

TI-Basic (TI-84 Plus CE), 100 bytes

Prompt X
{1→M
For(A,2,X
LM→L
A→dim(M
For(B,2,A–1
If A/2=int(A/2
Then
LL(B–1)LL(B→LM(B
Else
LL(B–1)+LL(B→LM(B
End
End
1→LM(dim(LM
End
LM

1-indexed, prompts user for input and prints a list containing the nth row of Pascal's Alternating Triangle.

While looping: _LM is the current row, and _LL is the previous row.

TI-Basic is a tokenized language. All tokens used here are one-byte tokens.

I think I can golf this further by modifying M in-place from the end.

Explanation:

Prompt X            # 3 bytes; get user input, store in X
{1→M                # 5 bytes, store the first row into LM
For(A,2,X           # 7 bytes, Loop X-1 times, with A as the counter, starting at 2
LM→L                # 5 bytes, copy list M into list L
A→dim(M             # 5 bytes, extend M by one
For(B,2,A–1         # 9 bytes, for each index B that isn't the first or last...
If A/2=int(A/2      # 10 bytes,    if A is even...
Then                # 2 bytes,     then...
LL(B–1)LL(B→LM(B     # 17 bytes,        the Bth item in this row is the Bth times the (B-1)th of the previous row
Else                # 2 bytes,     else...
LL(B–1)+LL(B→LM(B    # 18 bytes,        the Bth item in this row is the Bth plus the (B-1)th of the previous row
End                 # 2 bytes,     endif
End                 # 2 bytes,  endfor
1→LM(dim(LM         # 9 bytes, the last item is always 1
End                 # 2 bytes, endfor
LM                  # 2 bytes, Implicitly print the final row

Answer 19

JavaScript (ES6), 71 69 66 bytes

f=n=>n?(p=f(n-1),[...p.map((v,i)=>i--?n%2?v*p[i]:v+p[i]:1),1]):[1]

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0-indexed.
-3 bytes by @Arnauld

f=n=>n?(p=f(n-1),[...p.map((v,i)=>i--?n%2?v*p[i]:v+p[i]:1),1]):[1]

for (var i = 0; i < 10; ++i) {
  console.log(JSON.stringify(f(i)));
}