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.

3, 3, 12, 3, 15, 105, 3, 18, 162, 1944, 3, 21, 231, 3465, 65835, 3, 24, 312, 5616, 129168, 3616704, 3, 27, 405, 8505, 229635, 7577955, 295540245, 3, 30, 510, 12240, 379440, 14418720, 648842400, 33739804800, 3, 33, 627, 16929, 592515, 25478145, 1299385395, 76663738305, 5136470466435

Offset

0,1

Comments

Row sums are {3, 15, 123, 2127, 69555, 3751827, 303356775, 34403458143, 5214459678387, 1018396843935195, 249088654250968899, ...}.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

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

Example

Triangle begins as:
  3;
  3, 12;
  3, 15, 105;
  3, 18, 162,  1944;
  3, 21, 231,  3465,  65835;
  3, 24, 312,  5616, 129168,  3616704;
  3, 27, 405,  8505, 229635,  7577955, 295540245;
  3, 30, 510, 12240, 379440, 14418720, 648842400, 33739804800;

Maple

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

Mathematica

T[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j, 0, k}]];
Table[T[n, k, 2], {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(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
(Magma) m:=2;
function T(n, k)
  if k eq 0 then return NthPrime(m);
  else return (&*[j*n + NthPrime(m): j in [0..k]]);
  end if; return T; end function;
[T(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 03 2019
(Sage)
def T(n, k):
    m=2
    if (k==0): return nth_prime(m)
    else: return product(j*n + nth_prime(m) for j in (0..k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 03 2019

Crossrefs

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

Cf. A000142, A001710, A001813, A008544, A008545.

Cf. A001147, A032031, A047056, A144739, A144758.

Sequence in context: A163372 A349922 A066437 * A065957 A303309 A268774

Adjacent sequences: A153267 A153268 A153269 * A153271 A153272 A153273

Keyword

nonn,tabl

Author

Roger L. Bagula, Dec 22 2008

Extensions

Edited by G. C. Greubel, Dec 03 2019

Status

approved