A153272 -id:A153272 - cp's OEIS frontend

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.

5, 5, 30, 5, 35, 315, 5, 40, 440, 6160, 5, 45, 585, 9945, 208845, 5, 50, 750, 15000, 375000, 11250000, 5, 55, 935, 21505, 623645, 21827575, 894930575, 5, 60, 1140, 29640, 978120, 39124800, 1838865600, 99298742400, 5, 65, 1365, 39585, 1464645, 65909025, 3493178325, 213083877825, 14702787569925

Offset: 0

Roger L. Bagula, Dec 22 2008

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

			Triangle begins as:
  5;
  5, 30;
  5, 35, 315;
  5, 40, 440,  6160;
  5, 45, 585,  9945, 208845;
  5, 50, 750, 15000, 375000, 11250000;
  5, 55, 935, 21505, 623645, 21827575, 894930575;

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

Cf. A153271 (m=2), this sequence (m=3), A153272 (m=4).
Sequences related to m values:
m=2: A001147, A001710, A008545, A032031, A047056, A144739, A144758.
m=3: A001720, A007696, A008543, A034000, A051577, A052562, A147585, A147625, A147629.

m:=3;
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

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[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j,0,k}]];
Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten

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

def T(n, k):
    m=3
    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

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

Edited by G. C. Greubel, Dec 03 2019

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.

Original entry on oeis.org

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

Roger L. Bagula, Dec 22 2008

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

			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;

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

Cf. this sequence (m=2), A153271 (m=3), A153272 (m=4).
Cf. A000142, A001710, A001813, A008544, A008545.
Cf. A001147, A032031, A047056, A144739, A144758.

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

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[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

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

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

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

Edited by G. C. Greubel, Dec 03 2019

cp's OEIS Frontend

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.

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