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A153270 Triangle T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 2, read by rows.

%I A153270 #11 Sep 08 2022 08:45:39
%S A153270 3,3,12,3,15,105,3,18,162,1944,3,21,231,3465,65835,3,24,312,5616,
%T A153270 129168,3616704,3,27,405,8505,229635,7577955,295540245,3,30,510,12240,
%U A153270 379440,14418720,648842400,33739804800,3,33,627,16929,592515,25478145,1299385395,76663738305,5136470466435
%N A153270 Triangle T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 2, read by rows.
%C A153270 Row sums are {3, 15, 123, 2127, 69555, 3751827, 303356775, 34403458143, 5214459678387, 1018396843935195, 249088654250968899, ...}.
%H A153270 G. C. Greubel, <a href="/A153270/b153270.txt">Rows n = 0..100 of triangle, flattened</a>
%F A153270 T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 2.
%e A153270 Triangle begins as:
%e A153270   3;
%e A153270   3, 12;
%e A153270   3, 15, 105;
%e A153270   3, 18, 162,  1944;
%e A153270   3, 21, 231,  3465,  65835;
%e A153270   3, 24, 312,  5616, 129168,  3616704;
%e A153270   3, 27, 405,  8505, 229635,  7577955, 295540245;
%e A153270   3, 30, 510, 12240, 379440, 14418720, 648842400, 33739804800;
%p A153270 m:=2; seq(seq(`if`(k=0, ithprime(m), mul(j*n + ithprime(m), j=0..k)), k=0..n), n=0..10); # _G. C. Greubel_, Dec 03 2019
%t A153270 T[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j,0,k}]];
%t A153270 Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten
%o A153270 (PARI) T(n,k) = my(m=2); if(k==0, prime(m), prod(j=0,k, j*n + prime(m)) ); \\ _G. C. Greubel_, Dec 03 2019
%o A153270 (Magma) m:=2;
%o A153270 function T(n,k)
%o A153270   if k eq 0 then return NthPrime(m);
%o A153270   else return (&*[j*n + NthPrime(m): j in [0..k]]);
%o A153270   end if; return T; end function;
%o A153270 [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 03 2019
%o A153270 (Sage)
%o A153270 def T(n, k):
%o A153270     m=2
%o A153270     if (k==0): return nth_prime(m)
%o A153270     else: return product(j*n + nth_prime(m) for j in (0..k))
%o A153270 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 03 2019
%Y A153270 Cf. this sequence (m=2), A153271 (m=3), A153272 (m=4).
%Y A153270 Cf. A000142, A001710, A001813, A008544, A008545.
%Y A153270 Cf. A001147, A032031, A047056, A144739, A144758.
%K A153270 nonn,tabl
%O A153270 0,1
%A A153270 _Roger L. Bagula_, Dec 22 2008
%E A153270 Edited by _G. C. Greubel_, Dec 03 2019