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%I #8 Sep 08 2022 08:45:39
%S 0,1,3,2,10,80,3,21,231,3465,4,36,504,9576,229824,5,55,935,21505,
%T 623645,21827575,6,78,1560,42120,1432080,58715280,2818333440,7,105,
%U 2415,74865,2919735,137227545,7547514975,475493443425,8,136,3536,123760,5445440,288608320,17893715840,1270453824640,101636305971200
%N Triangle sequence: T(n, k) = -Product_{j=0..k+1} ((n+1)*j - 1).
%C Row sums are: {0, 4, 92, 3720, 239944, 22473720, 2878524564, 483181183072, 102924947692880, 27128289837188700, ...}.
%H G. C. Greubel, <a href="/A153187/b153187.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n, k) = -Product_{j=0..k+1} (j*(n+1) - 1).
%F T(n, k) = -(n+1)^(k+2) * Pochhammer(-1/(n+1), k+2).
%e Triangle begins as:
%e 0;
%e 1, 3;
%e 2, 10, 80;
%e 3, 21, 231, 3465;
%e 4, 36, 504, 9576, 229824;
%e 5, 55, 935, 21505, 623645, 21827575;
%e 6, 78, 1560, 42120, 1432080, 58715280, 2818333440;
%e 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, 475493443425;
%p seq(seq(-mul(j*(n+1)-1, j = 0..k+1), k = 0..n), n = 0..10); # _G. C. Greubel_, Mar 05 2020
%t T[n_, m_] = -Product[(n+1)*j -1, {j,0,m+1}]; Table[T[n, m], {n,0,10}, {m,0,n}]//Flatten
%t Table[-(n+1)^(k+2)*Pochhammer[-1/(n+1), k+2], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 05 2020 *)
%o (PARI) T(n,k) = (-1)*prod(j=0, k+1, j*(n+1)-1);
%o for(j=0, 10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Mar 05 2020
%o (Magma) [-(&*[j*(n+1)-1: j in [0..k+1]]): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Mar 05 2020
%o (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
%o (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
%Y Cf. A001477, A014105, A033593, A033594, A153273.
%K nonn,tabl
%O 0,3
%A _Roger L. Bagula_, Dec 20 2008
%E Edited by _G. C. Greubel_, Mar 05 2020
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