A317644 Triangle read by rows: multiplicative version of Pascal's triangle except n-th row begins and ends with (n+1)-st prime.

2, 3, 3, 5, 9, 5, 7, 45, 45, 7, 11, 315, 2025, 315, 11, 13, 3465, 637875, 637875, 3465, 13, 17, 45045, 2210236875, 406884515625, 2210236875, 45045, 17, 19, 765765, 99560120034375, 899311160300888671875, 899311160300888671875, 99560120034375, 765765, 19, 23, 14549535, 76239655318123171875, 89535527067809533413858673095703125, 808760563041730681160065242862701416015625, 89535527067809533413858673095703125, 76239655318123171875, 14549535, 23

Offset: 0

Author

Philipp O. Tsvetkov, Aug 02 2018

Keywords

• nonn
• tabl

Examples

			Triangle begins:
   2;
   3,      3;
   5,      9,      5;
   7,     45,     45,      7;
  11,    315,   2025,    315,     11;
  13,   3465, 637875, 637875,   3465,     13;
  ...
Formatted as a symmetric triangle:
.
                       2
.
                   3       3
.
               5       9       5
.
           7      45      45       7
.
      11      315    2025     315     11
.
  13     3465   637875  637875   3465     13
...
		

Crossrefs

Cf. A000040, A007318, A007949, A028326, A051599, A080046, A112765.

Programs

Mathematica

t = {{2}};
Table[AppendTo[
    t, {Prime[i],
      Table[
       t[[i - 1]][[j]]*t[[i - 1]][[j + 1]], {j,
        1, (t[[i - 1]] // Length) - 1}], Prime[i]} // Flatten], {i, 2, 10}] //
   Last // Flatten
t={}; Do[r={}; Do[If[k==0||k==n, m=Prime[n + 1], m=t[[n, k]]t[[n, k + 1]]]; r=AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t (* Vincenzo Librandi, Sep 03 2018 *)

Formula

From Rémy Sigrist, Sep 02 2018: (Start)

A007949(T(n+1, k+1)) = A028326(n, k) for any n >= 0 and k = 0..n.

A112765(T(n+1, k+1)) = A007318(n, k) for any n > 0 and k = 0..n.

(End)