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%I #12 Sep 08 2022 08:45:50
%S 1,1,1,1,-1,1,1,-17,-17,1,1,-293,-309,-293,1,1,-9449,-9741,-9741,
%T -9449,1,1,-605429,-614877,-615153,-614877,-605429,1,1,-77514569,
%U -78119997,-78129429,-78129429,-78119997,-77514569,1,1,-19844960309,-19922474877,-19923080289,-19923089445,-19923080289,-19922474877,-19844960309,1
%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 = 2.
%C Row sums are: {1, 2, 1, -32, -893, -38378, -3055763, -467527988, -139304120393,
%C -81405588536318, -93713294552041343, ...}.
%H G. C. Greubel, <a href="/A172091/b172091.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=2.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, -1, 1;
%e 1, -17, -17, 1;
%e 1, -293, -309, -293, 1;
%e 1, -9449, -9741, -9741, -9449, 1;
%e 1, -605429, -614877, -615153, -614877, -605429, 1;
%p T:= proc(n, k, q) option remember;
%p c(n,q):= mul( 2^(j+1) -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,2), k=0..n), n=0..10); # _G. C. Greubel_, Dec 05 2019
%t c[n_, q_]:= Product[2^(j+1) -1, {j, n-1}]; T[n_, k_, q_]:= c[k, q] + c[n-k, q] - c[n, q]; Table[T[n, k, 2], {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, 2^(j+1) -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,2), ", "))) \\ _G. C. Greubel_, Dec 05 2019
%o (Magma) c:= func< n,q | n lt 2 select 1 else &*[2^(j+1) -1: j in [1..n-1]] >;
%o [c(k,2) + c(n-k,2) - c(n,2): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 05 2019
%o (Sage)
%o def c(n,q): return product( 2^(j+1) - 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,2) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 05 2019
%Y Cf. this sequence (q=2), A172092 (q=3), A172093 (q=4).
%K sign,tabl
%O 0,8
%A _Roger L. Bagula_, Jan 25 2010
%E Edited by _G. C. Greubel_, Dec 05 2019
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