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A153271 Triangle T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 3, read by rows.

%I A153271 #12 Sep 08 2022 08:45:39
%S A153271 5,5,30,5,35,315,5,40,440,6160,5,45,585,9945,208845,5,50,750,15000,
%T A153271 375000,11250000,5,55,935,21505,623645,21827575,894930575,5,60,1140,
%U A153271 29640,978120,39124800,1838865600,99298742400,5,65,1365,39585,1464645,65909025,3493178325,213083877825,14702787569925
%N A153271 Triangle T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 3, read by rows.
%C A153271 Row sums are {5, 35, 355, 6645, 219425, 11640805, 917404295, 101177741765, 14919432040765, 2839006665525525, 677815000136926955, ...}.
%H A153271 G. C. Greubel, <a href="/A153271/b153271.txt">Rows n = 0..100 of triangle, flattened</a>
%F A153271 T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 3.
%e A153271 Triangle begins as:
%e A153271   5;
%e A153271   5, 30;
%e A153271   5, 35, 315;
%e A153271   5, 40, 440,  6160;
%e A153271   5, 45, 585,  9945, 208845;
%e A153271   5, 50, 750, 15000, 375000, 11250000;
%e A153271   5, 55, 935, 21505, 623645, 21827575, 894930575;
%p A153271 m:=3; 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 A153271 T[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j,0,k}]];
%t A153271 Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten
%o A153271 (PARI) T(n,k) = my(m=3); if(k==0, prime(m), prod(j=0,k, j*n + prime(m)) ); \\ _G. C. Greubel_, Dec 03 2019
%o A153271 (Magma) m:=3;
%o A153271 function T(n,k)
%o A153271   if k eq 0 then return NthPrime(m);
%o A153271   else return (&*[j*n + NthPrime(m): j in [0..k]]);
%o A153271   end if; return T; end function;
%o A153271 [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 03 2019
%o A153271 (Sage)
%o A153271 def T(n, k):
%o A153271     m=3
%o A153271     if (k==0): return nth_prime(m)
%o A153271     else: return product(j*n + nth_prime(m) for j in (0..k))
%o A153271 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 03 2019
%Y A153271 Cf. A153271 (m=2), this sequence (m=3), A153272 (m=4).
%Y A153271 Sequences related to m values:
%Y A153271 m=2: A001147, A001710, A008545, A032031, A047056, A144739, A144758.
%Y A153271 m=3: A001720, A007696, A008543, A034000, A051577, A052562, A147585, A147625, A147629.
%K A153271 nonn,tabl
%O A153271 0,1
%A A153271 _Roger L. Bagula_, Dec 22 2008
%E A153271 Edited by _G. C. Greubel_, Dec 03 2019