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A174719 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 3, read by rows.

%I A174719 #9 Feb 09 2021 21:40:00
%S A174719 1,1,1,1,-7,1,1,-51,-51,1,1,-239,-399,-239,1,1,-967,-2177,-2177,-967,
%T A174719 1,1,-3639,-10191,-13831,-10191,-3639,1,1,-13115,-43719,-74323,-74323,
%U A174719 -43719,-13115,1,1,-45919,-177119,-360799,-452639,-360799,-177119,-45919,1
%N A174719 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 3, read by rows.
%C A174719 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 A174719 G. C. Greubel, <a href="/A174719/b174719.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A174719 T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q=3.
%F A174719 Sum_{k=0..n} T(n, k, 3) = 3^n*(n+1) + 2^n*(1 - 3^n) = A027471(n+2) - A248216(n). - _G. C. Greubel_, Feb 09 2021
%e A174719 Triangle begins as:
%e A174719   1;
%e A174719   1,      1;
%e A174719   1,     -7,       1;
%e A174719   1,    -51,     -51,       1;
%e A174719   1,   -239,    -399,    -239,       1;
%e A174719   1,   -967,   -2177,   -2177,    -967,       1;
%e A174719   1,  -3639,  -10191,  -13831,  -10191,   -3639,       1;
%e A174719   1, -13115,  -43719,  -74323,  -74323,  -43719,  -13115,      1;
%e A174719   1, -45919, -177119, -360799, -452639, -360799, -177119, -45919, 1;
%t A174719 T[n_, k_, q_]:= 1 +(1-q^n)*(Binomial[n, k] -1);
%t A174719 Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten
%o A174719 (Sage)
%o A174719 def T(n,k,q): return 1 + (1-q^n)*(binomial(n,k) - 1)
%o A174719 flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 09 2021
%o A174719 (Magma)
%o A174719 T:= func< n,k,q | 1 + (1-q^n)*(Binomial(n,k) -1) >;
%o A174719 [T(n,k,3): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 09 2021
%Y A174719 Cf. A000012 (q=1), A174718 (q=2), this sequence (q=3), A174720 (q=4).
%Y A174719 Cf. A027471, A248216.
%K A174719 sign,tabl
%O A174719 0,5
%A A174719 _Roger L. Bagula_, Mar 28 2010
%E A174719 Edited by _G. C. Greubel_, Feb 09 2021