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A174718 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 2, read by rows.

%I A174718 #9 Feb 09 2021 21:37:34
%S A174718 1,1,1,1,-2,1,1,-13,-13,1,1,-44,-74,-44,1,1,-123,-278,-278,-123,1,1,
%T A174718 -314,-881,-1196,-881,-314,1,1,-761,-2539,-4317,-4317,-2539,-761,1,1,
%U A174718 -1784,-6884,-14024,-17594,-14024,-6884,-1784,1,1,-4087,-17884,-42412,-63874,-63874,-42412,-17884,-4087,1
%N A174718 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 2, read by rows.
%C A174718 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 A174718 G. C. Greubel, <a href="/A174718/b174718.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A174718 T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q=2.
%F A174718 Sum_{k=0..n} T(n, k, 2) = 2^n *(n + 2 - 2^n) = A001787(n+1) - A020522(n). - _G. C. Greubel_, Feb 09 2021
%e A174718 Triangle begins as:
%e A174718   1;
%e A174718   1,     1;
%e A174718   1,    -2,      1;
%e A174718   1,   -13,    -13,      1;
%e A174718   1,   -44,    -74,    -44,      1;
%e A174718   1,  -123,   -278,   -278,   -123,      1;
%e A174718   1,  -314,   -881,  -1196,   -881,   -314,      1;
%e A174718   1,  -761,  -2539,  -4317,  -4317,  -2539,   -761,      1;
%e A174718   1, -1784,  -6884, -14024, -17594, -14024,  -6884,  -1784,     1;
%e A174718   1, -4087, -17884, -42412, -63874, -63874, -42412, -17884, -4087, 1;
%t A174718 T[n_, k_, q_]:= 1 +(1-q^n)*(Binomial[n, k] -1);
%t A174718 Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten
%o A174718 (Sage)
%o A174718 def T(n,k,q): return 1 + (1-q^n)*(binomial(n,k) - 1)
%o A174718 flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 09 2021
%o A174718 (Magma)
%o A174718 T:= func< n,k,q | 1 + (1-q^n)*(Binomial(n,k) -1) >;
%o A174718 [T(n,k,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 09 2021
%Y A174718 Cf. A000012 (q=1), this sequence (q=2), A174719 (q=3), A174720 (q=4).
%Y A174718 Cf. A001787, A020522.
%K A174718 sign,tabl
%O A174718 0,5
%A A174718 _Roger L. Bagula_, Mar 28 2010
%E A174718 Edited by _G. C. Greubel_, Feb 09 2021