%I #7 Sep 08 2022 08:45:50
%S 1,1,1,1,-2,1,1,-47,-47,1,1,-2027,-2072,-2027,1,1,-249599,-251624,
%T -251624,-249599,1,1,-91359839,-91609436,-91611416,-91609436,
%U -91359839,1,1,-100039779839,-100131139676,-100131389228,-100131389228,-100131139676,-100039779839,1
%N Triangle, read by rows, T(n,k,q) = c(k,q) + c(n-k,q) - c(n, q) where c(n,q) = Product_{j=1..n-1} ((q^(j+1) - 1)/(q-1)) and q = 3.
%C Row sums are: {1, 2, 0, -92, -6124, -1002444, -457549964, -600604617484, -2298816299112204, -25856055844713627404, -858811326017167374184204, ...}.
%H G. C. Greubel, <a href="/A172092/b172092.txt">Rows n = 0..50 of triangle, flattened</a>
%F Let c(n,q) = Product_{j=1..n-1} ((q^(j+1) - 1)/(q-1)) then T(n,k,q) = -c(n,q) + c(n-k,q) + c(k, q) for q=3.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, -2, 1;
%e 1, -47, -47, 1;
%e 1, -2027, -2072, -2027, 1;
%e 1, -249599, -251624, -251624, -249599, 1;
%e 1, -91359839, -91609436, -91611416, -91609436, -91359839, 1;
%p T:= proc(n, k, q) option remember;
%p c(n,q):= mul( (q^(j+1) -1)/(q-1), j=1..n-1);
%p T(n,k,q):= c(k,q) + c(n-k,q) - c(n,q);
%p end:
%p seq(seq(T(n,k,3), k=0..n), n=0..10); # _G. C. Greubel_, Dec 05 2019
%t c[n_, q_]:= Product[(q^(j+1) -1)/(q-1), {j, n-1}]; T[n_, k_, q_]:= c[k, q] + c[n-k, q] - c[n, q]; Table[T[n, k, 3], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Dec 05 2019 *)
%o (PARI) c(n,q) = prod(j=1, n-1, (q^(j+1) -1)/(q-1));
%o T(n, k, q) = c(k,q) + c(n-k,q) - c(n,q);
%o for(n=0, 10, for(k=0, n, print1(T(n,k,3), ", "))) \\ _G. C. Greubel_, Dec 05 2019
%o (Magma) c:= func< n,q | n lt 2 select 1 else &*[(q^(j+1) -1)/(q-1): j in [1..n-1]] >;
%o T:= func< n,k,q | c(k,q) + c(n-k,q) - c(n,q) >;
%o [T(n,k,3): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 05 2019
%o (Sage)
%o def c(n,q): return product( (q^(j+1) -1)/(q-1) for j in (1..n-1))
%o def T(n,k,q): return c(k,q) + c(n-k,q) - c(n,q)
%o [[T(n,k,3) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 05 2019
%Y Cf. A172091 (q=2), this sequence (q=3), A172093 (q=4).
%K sign,tabl
%O 0,5
%A _Roger L. Bagula_, Jan 25 2010
%E Edited by _G. C. Greubel_, Dec 05 2019
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