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%I #7 Feb 09 2021 21:40:07
%S 1,1,1,1,-14,1,1,-125,-125,1,1,-764,-1274,-764,1,1,-4091,-9206,-9206,
%T -4091,1,1,-20474,-57329,-77804,-57329,-20474,1,1,-98297,-327659,
%U -557021,-557021,-327659,-98297,1,1,-458744,-1769444,-3604424,-4521914,-3604424,-1769444,-458744,1
%N Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 4, read by rows.
%C The row sums of this class of sequences, for varying q, is given by Sum_{k=0..n} T(n, k, q) = q^n * (n+1) + 2^n * (1 - q^n). - _G. C. Greubel_, Feb 09 2021
%H G. C. Greubel, <a href="/A174720/b174720.txt">Rows n = 0..100 of the triangle, flattened</a>
%F T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q=4.
%F Sum_{k=0..n} T(n, k, 4) = 4^n*(n+1) + 2^n*(1 - 4^n) = A002697(n+1) - A248217(n). - _G. C. Greubel_, Feb 09 2021
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, -14, 1;
%e 1, -125, -125, 1;
%e 1, -764, -1274, -764, 1;
%e 1, -4091, -9206, -9206, -4091, 1;
%e 1, -20474, -57329, -77804, -57329, -20474, 1;
%e 1, -98297, -327659, -557021, -557021, -327659, -98297, 1;
%e 1, -458744, -1769444, -3604424, -4521914, -3604424, -1769444, -458744, 1;
%t T[n_, k_, q_]:= 1 +(1-q^n)*(Binomial[n, k] -1);
%t Table[T[n,k,4], {n,0,12}, {k,0,n}]//Flatten
%o (Sage)
%o def T(n,k,q): return 1 + (1-q^n)*(binomial(n,k) - 1)
%o flatten([[T(n,k,4) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 09 2021
%o (Magma)
%o T:= func< n,k,q | 1 + (1-q^n)*(Binomial(n,k) -1) >;
%o [T(n,k,4): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 09 2021
%Y Cf. A000012 (q=1), A174718 (q=2), A174719 (q=3), this sequence (q=4).
%Y Cf. A002697, A248217.
%K sign,tabl
%O 0,5
%A _Roger L. Bagula_, Mar 28 2010
%E Edited by _G. C. Greubel_, Feb 09 2021
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