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
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Dec 20 2008
EXTENSIONS
Edited by G. C. Greubel, Mar 05 2020
STATUS
approved