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

%I A317644 #21 Sep 23 2018 23:02:49
%S A317644 2,3,3,5,9,5,7,45,45,7,11,315,2025,315,11,13,3465,637875,637875,3465,
%T A317644 13,17,45045,2210236875,406884515625,2210236875,45045,17,19,765765,
%U A317644 99560120034375,899311160300888671875,899311160300888671875,99560120034375,765765,19,23,14549535,76239655318123171875,89535527067809533413858673095703125,808760563041730681160065242862701416015625,89535527067809533413858673095703125,76239655318123171875,14549535,23
%N A317644 Triangle read by rows: multiplicative version of Pascal's triangle except n-th row begins and ends with (n+1)-st prime.
%F A317644 From _Rémy Sigrist_, Sep 02 2018: (Start)
%F A317644 A007949(T(n+1, k+1)) = A028326(n, k) for any n >= 0 and k = 0..n.
%F A317644 A112765(T(n+1, k+1)) = A007318(n, k) for any n > 0 and k = 0..n.
%F A317644 (End)
%e A317644 Triangle begins:
%e A317644    2;
%e A317644    3,      3;
%e A317644    5,      9,      5;
%e A317644    7,     45,     45,      7;
%e A317644   11,    315,   2025,    315,     11;
%e A317644   13,   3465, 637875, 637875,   3465,     13;
%e A317644   ...
%e A317644 Formatted as a symmetric triangle:
%e A317644 .
%e A317644                        2
%e A317644 .
%e A317644                    3       3
%e A317644 .
%e A317644                5       9       5
%e A317644 .
%e A317644            7      45      45       7
%e A317644 .
%e A317644       11      315    2025     315     11
%e A317644 .
%e A317644   13     3465   637875  637875   3465     13
%e A317644 ...
%t A317644 t = {{2}};
%t A317644 Table[AppendTo[
%t A317644     t, {Prime[i],
%t A317644       Table[
%t A317644        t[[i - 1]][[j]]*t[[i - 1]][[j + 1]], {j,
%t A317644         1, (t[[i - 1]] // Length) - 1}], Prime[i]} // Flatten], {i, 2, 10}] //
%t A317644    Last // Flatten
%t A317644 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 *)
%Y A317644 Cf. A000040, A007318, A007949, A028326, A051599, A080046, A112765.
%K A317644 nonn,tabl
%O A317644 0,1
%A A317644 _Philipp O. Tsvetkov_, Aug 02 2018