A153274 Triangle, read by rows, T(n,k) = k^(n+1) * Pochhammer(1/k, n+1).

2, 6, 15, 24, 105, 280, 120, 945, 3640, 9945, 720, 10395, 58240, 208845, 576576, 5040, 135135, 1106560, 5221125, 17873856, 49579075, 40320, 2027025, 24344320, 151412625, 643458816, 2131900225, 5925744000, 362880, 34459425, 608608000, 4996616625, 26381811456, 104463111025, 337767408000, 939536222625

Offset

1,1

Comments

A Pochhammer function-based triangular sequence.

Row sums are: {2, 21, 409, 14650, 854776, 73920791, 8878927331, 1413788600036, 288152651134776, 73152069870215127, ...}.

Links

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Formula

T(n, k) = k^(n+1) * Pochmammer(1/k, n+1).

T(n, k) = Product_{j=0..n} (j*k + 1). - G. C. Greubel, Mar 05 2020

Example

Triangle begins as:
      2;
      6,      15;
     24,     105,      280;
    120,     945,     3640,      9945;
    720,   10395,    58240,    208845,    576576;
   5040,  135135,  1106560,   5221125,  17873856,   49579075;
  40320, 2027025, 24344320, 151412625, 643458816, 2131900225, 5925744000;

Maple

seq(seq( k^(n+1)*pochhammer(1/k, n+1), k=1..n), n=1..12); # G. C. Greubel, Mar 05 2020

Mathematica

Table[Apply[Plus, CoefficientList[j*k^n*Pochhammer[(j+k)/k, n], j]], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 05 2020 *)
Table[k^(n+1)*Pochhammer[1/k, n+1], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Mar 05 2020 *)

Prog

(PARI) T(n, k) = prod(j=0, n, j*k+1);
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Mar 05 2020
(Magma) [(&*[j*k+1: j in [0..n]]): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 05 2020
(Sage) [[k^(n+1)*rising_factorial(1/k, n+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 05 2020
(GAP) Flat(List([1..12], n-> List([1..n], k-> Product([0..n], j-> j*k+1 )))); # G. C. Greubel, Mar 05 2020

Crossrefs

Cf. A000142, A001147, A007559, A007696, A008548, A008542, A045754, A045755, A045756.

Cf. A142589, A256268.

Sequence in context: A090979 A050508 A033298 * A091766 A269706 A293402

Adjacent sequences: A153271 A153272 A153273 * A153275 A153276 A153277

Keyword

nonn,tabl

Author

Roger L. Bagula, Dec 22 2008

Extensions

Edited by G. C. Greubel, Mar 05 2020

Status

approved