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A176263 Triangle T(n,k) = A015440(k) - A015440(n) + A015440(n-k), read by rows.

%I A176263 #14 Jun 29 2025 18:28:36
%S A176263 1,1,1,1,-4,1,1,-4,-4,1,1,-29,-29,-29,1,1,-54,-79,-79,-54,1,1,-204,
%T A176263 -254,-279,-254,-204,1,1,-479,-679,-729,-729,-679,-479,1,1,-1504,
%U A176263 -1979,-2179,-2204,-2179,-1979,-1504,1,1,-3904,-5404,-5879,-6054,-6054,-5879,-5404,-3904,1
%N A176263 Triangle T(n,k) = A015440(k) - A015440(n) + A015440(n-k), read by rows.
%C A176263 Row sums are s(n) = {1, 2, -2, -6, -85, -264, -1193, -3772, -13526, -42480, -139159, ...}, obeying s(n) = 3*s(n-1) + 7*s(n-2) - 19*s(n-3) - 15*s(n-4) + 25*s(n-5) with g.f. (1-x-15*x^2+5*x^3)/((1-x)*(1-x-5*x^2)^2).
%H A176263 G. C. Greubel, <a href="/A176263/b176263.txt">Rows n = 0..100 of triangle, flattened</a>
%e A176263 Triangle begins as:
%e A176263   1;
%e A176263   1,     1;
%e A176263   1,    -4,     1;
%e A176263   1,    -4,    -4,     1;
%e A176263   1,   -29,   -29,   -29,     1;
%e A176263   1,   -54,   -79,   -79,   -54,     1;
%e A176263   1,  -204,  -254,  -279,  -254,  -204,     1;
%e A176263   1,  -479,  -679,  -729,  -729,  -679,  -479,     1;
%e A176263   1, -1504, -1979, -2179, -2204, -2179, -1979, -1504,     1;
%e A176263   1, -3904, -5404, -5879, -6054, -6054, -5879, -5404, -3904, 1;
%p A176263 A176263 := proc(n,k)
%p A176263         A015440(k)-A015440(n)+A015440(n-k) ;
%p A176263 end proc; # _R. J. Mathar_, May 03 2013
%t A176263 (* Set of sequences q=0..10. This sequence is q=5. *)
%t A176263 f[n_, q_]:= f[n, q] = If[n<2, n, f[n-1, q] + q*f[n-2, q]];
%t A176263 T[n_, k_, q_]:= f[k+1, q] + f[n-k+1, q] - f[n+1, q];
%t A176263 Table[Flatten[Table[T[n, k, q], {n,0,10}, {k,0,n}]], {q,0,10}]
%t A176263 (* Second program *)
%t A176263 A015440[n_]:= Sum[5^j*Binomial[n-j, j], {j,0,(n+1)/2}]; T[n_, k_]:= T[n, k]= A015440[k] +A015440[n-k] -A015440[n]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 24 2019 *)
%o A176263 (PARI) A015440(n) = sum(j=0,(n+1)\2, 5^j*binomial(n-j,j));
%o A176263 T(n,k) = A015440(k) - A015440(n) + A015440(n-k); \\ _G. C. Greubel_, Nov 24 2019
%o A176263 (Magma) A015440:= func< n | &+[5^j*Binomial(n-j,j): j in [0..Floor(n/2)]] >;
%o A176263 [A015440(k) - A015440(n) + A015440(n-k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 24 2019
%o A176263 (Sage)
%o A176263 def A015440(n): return sum(5^j*binomial(n-j,j) for j in (0..floor(n/2)))
%o A176263 [[A015440(k) - A015440(n) + A015440(n-k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 24 2019
%K A176263 sign,tabl,easy
%O A176263 0,5
%A A176263 _Roger L. Bagula_, Apr 13 2010