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%I #16 Jun 10 2024 08:50:27
%S -1,-1,1,-1,2,8,-1,3,11,23,-1,4,14,29,49,-1,5,17,35,59,89,-1,6,20,41,
%T 69,104,146,-1,7,23,47,79,119,167,223,-1,8,26,53,89,134,188,251,323,
%U -1,9,29,59,99,149,209,279,359,449,-1,10,32,65,109,164,230,307,395,494,604
%N Triangle read by rows T(n, k) = n + (n+1)*(k-1)*(k+2)/2.
%H G. C. Greubel, <a href="/A143198/b143198.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = (n+1)*A000096(k-1) + n, for 0 <= k <= n, n >= 0.
%F Sum_{k=0..n} T(n, k) = (n-1)*(n+1)*(n^2+4*n+6)/6 (row sums).
%F Sum_{k=0..n} (-1)^k*T(n, k) = ((-1)^n*(2*n^3+6*n^2+5*n-3) -n-5)/8. - _G. C. Greubel_, Jun 09 2024
%e Triangle begins as:
%e -1;
%e -1, 1;
%e -1, 2, 8;
%e -1, 3, 11, 23;
%e -1, 4, 14, 29, 49;
%e -1, 5, 17, 35, 59, 89;
%e -1, 6, 20, 41, 69, 104, 146;
%e -1, 7, 23, 47, 79, 119, 167, 223;
%e -1, 8, 26, 53, 89, 134, 188, 251, 323;
%e -1, 9, 29, 59, 99, 149, 209, 279, 359, 449;
%e -1, 10, 32, 65, 109, 164, 230, 307, 395, 494, 604;
%p A143198 := proc(n,m)
%p n+(n+1)*(m-1)*(m+2)/2 ;
%p end proc:
%p seq(seq(A143198(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Nov 11 2011
%t Table[n+(n+1)*(k-1)*(k+2)/2, {n,0,15}, {k,0,n}]//Flatten
%o (Magma) [n+(n+1)*(k-1)*(k+2)/2: k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jun 09 2024
%o (SageMath) flatten([[n+(n+1)*(k-1)*(k+2)/2 for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Jun 09 2024
%Y Cf. A000096.
%K easy,sign,tabl
%O 0,5
%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 20 2008
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