A153187 Triangle sequence: T(n, k) = -Product_{j=0..k+1} ((n+1)*j - 1).

0, 1, 3, 2, 10, 80, 3, 21, 231, 3465, 4, 36, 504, 9576, 229824, 5, 55, 935, 21505, 623645, 21827575, 6, 78, 1560, 42120, 1432080, 58715280, 2818333440, 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, 475493443425, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840, 1270453824640, 101636305971200

Offset

0,3

Comments

Row sums are: {0, 4, 92, 3720, 239944, 22473720, 2878524564, 483181183072, 102924947692880, 27128289837188700, ...}.

Links

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

Formula

T(n, k) = -Product_{j=0..k+1} (j*(n+1) - 1).

T(n, k) = -(n+1)^(k+2) * Pochhammer(-1/(n+1), k+2).

Example

Triangle begins as:
  0;
  1,   3;
  2,  10,   80;
  3,  21,  231,  3465;
  4,  36,  504,  9576,  229824;
  5,  55,  935, 21505,  623645,  21827575;
  6,  78, 1560, 42120, 1432080,  58715280,  2818333440;
  7, 105, 2415, 74865, 2919735, 137227545,  7547514975, 475493443425;

Maple

seq(seq(-mul(j*(n+1)-1, j = 0..k+1), k = 0..n), n = 0..10); # G. C. Greubel, Mar 05 2020

Mathematica

T[n_, m_] = -Product[(n+1)*j -1, {j, 0, m+1}]; Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten
Table[-(n+1)^(k+2)*Pochhammer[-1/(n+1), k+2], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 05 2020 *)

Prog

(PARI) T(n, k) = (-1)*prod(j=0, k+1, j*(n+1)-1);
for(j=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Mar 05 2020
(Magma) [-(&*[j*(n+1)-1: j in [0..k+1]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 05 2020
(Sage) [[-(n+1)^(k+2)*rising_factorial(-1/(n+1), k+2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 05 2020
(GAP) Flat(List([0..10], n-> List([0..n], k-> (-1)*Product([0..k+1], j-> j*(n+1) -1) ))); # G. C. Greubel, Mar 05 2020

Crossrefs

Cf. A001477, A014105, A033593, A033594, A153273.

Sequence in context: A096073 A088302 A074199 * A248242 A215598 A322603

Adjacent sequences: A153184 A153185 A153186 * A153188 A153189 A153190

Keyword

nonn,tabl

Author

Roger L. Bagula, Dec 20 2008

Extensions

Edited by G. C. Greubel, Mar 05 2020

Status

approved