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A174720 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 4, read by rows.

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