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A176261 Triangle T(n,k) = A006130(k) - A006130(n) + A006130(n-k) read by rows.

%I A176261 #15 Sep 08 2022 08:45:52
%S A176261 1,1,1,1,-2,1,1,-2,-2,1,1,-11,-11,-11,1,1,-20,-29,-29,-20,1,1,-56,-74,
%T A176261 -83,-74,-56,1,1,-119,-173,-191,-191,-173,-119,1,1,-290,-407,-461,
%U A176261 -470,-461,-407,-290,1,1,-650,-938,-1055,-1100,-1100,-1055,-938,-650,1
%N A176261 Triangle T(n,k) = A006130(k) - A006130(n) + A006130(n-k) read by rows.
%C A176261 Row sums are s(n) = {1, 2, 0, -2, -31, -96, -341, -964, -2784, -7484, -20041, ...}, obey s(n) = 3*s(n-1) + 3*s(n-2) - 11*s(n-3) - 3*s(n-4) + 9*s(n-5) and have g.f. (1-x+3*x^3-9*x^2)/((1-x)*(1-x-3*x^2)^2).
%H A176261 G. C. Greubel, <a href="/A176261/b176261.txt">Rows n = 0..100 of triangle, flattened</a>
%F A176261 T(n,k) = T(n,n-k).
%F A176261 T(n,k) = A006130(k) - A006130(n) + A006130(n-k), where A006130(n) = Sum_{j=0..n} binomial(n-j, j)*3^j. - _G. C. Greubel_, Nov 24 2019
%e A176261 Triangle begins as:
%e A176261   1;
%e A176261   1,     1;
%e A176261   1,    -2,     1;
%e A176261   1,    -2,    -2,     1;
%e A176261   1,   -11,   -11,   -11,     1;
%e A176261   1,   -20,   -29,   -29,   -20,     1;
%e A176261   1,   -56,   -74,   -83,   -74,   -56,     1;
%e A176261   1,  -119,  -173,  -191,  -191,  -173,  -119,     1;
%e A176261   1,  -290,  -407,  -461,  -470,  -461,  -407,  -290,     1;
%e A176261   1,  -650,  -938, -1055, -1100, -1100, -1055,  -938,  -650,     1;
%e A176261   1, -1523, -2171, -2459, -2567, -2603, -2567, -2459, -2171, -1523, 1;
%p A176261 A176261 := proc(n,k)
%p A176261         A006130(k)-A006130(n)+A006130(n-k) ;
%p A176261 end proc; # _R. J. Mathar_, May 03 2013
%t A176261 A006130[n_]:= Sum[Binomial[n-j,j]*3^j, {j,0,n}]; T[n_,k_]:= A006130[k] - A006130[n] + A006130[n-k]; Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Nov 24 2019 *)
%o A176261 (PARI) A006130(n) = sum(j=0,n,binomial(n-j,j)*3^j);
%o A176261 T(n,k) = A006130(k) -A006130(n) +A006130(n-k); \\ _G. C. Greubel_, Nov 24 2019
%o A176261 (Magma) A006130:= func< n | &+[Binomial(n-j,j)*3^j: j in [0..n]] >;
%o A176261 [A006130(k) -A006130(n) +A006130(n-k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 24 2019
%o A176261 (Sage)
%o A176261 def A006130(n): return sum(binomial(n-j,j)*3^j for j in (0..n))
%o A176261 [[A006130(k) -A006130(n) +A006130(n-k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 24 2019
%Y A176261 Cf. A006130.
%K A176261 sign,tabl,easy
%O A176261 0,5
%A A176261 _Roger L. Bagula_, Apr 13 2010