A153271 - OEIS

%I #12 Sep 08 2022 08:45:39

%S 5,5,30,5,35,315,5,40,440,6160,5,45,585,9945,208845,5,50,750,15000,

%T 375000,11250000,5,55,935,21505,623645,21827575,894930575,5,60,1140,

%U 29640,978120,39124800,1838865600,99298742400,5,65,1365,39585,1464645,65909025,3493178325,213083877825,14702787569925

%N 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 Row sums are {5, 35, 355, 6645, 219425, 11640805, 917404295, 101177741765, 14919432040765, 2839006665525525, 677815000136926955, ...}.

%H G. C. Greubel, <a href="/A153271/b153271.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 3.

%e Triangle begins as:

%e   5;

%e   5, 30;

%e   5, 35, 315;

%e   5, 40, 440,  6160;

%e   5, 45, 585,  9945, 208845;

%e   5, 50, 750, 15000, 375000, 11250000;

%e   5, 55, 935, 21505, 623645, 21827575, 894930575;

%p 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 T[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j,0,k}]];

%t Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten

%o (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 (Magma) m:=3;

%o function T(n,k)

%o   if k eq 0 then return NthPrime(m);

%o   else return (&*[j*n + NthPrime(m): j in [0..k]]);

%o   end if; return T; end function;

%o [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 03 2019

%o (Sage)

%o def T(n, k):

%o     m=3

%o     if (k==0): return nth_prime(m)

%o     else: return product(j*n + nth_prime(m) for j in (0..k))

%o [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 03 2019

%Y Cf. A153271 (m=2), this sequence (m=3), A153272 (m=4).

%Y Sequences related to m values:

%Y m=2: A001147, A001710, A008545, A032031, A047056, A144739, A144758.

%Y m=3: A001720, A007696, A008543, A034000, A051577, A052562, A147585, A147625, A147629.

%K nonn,tabl

%O 0,1

%A _Roger L. Bagula_, Dec 22 2008

%E Edited by _G. C. Greubel_, Dec 03 2019