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%I #11 Sep 08 2022 08:45:52
%S 1,1,1,1,-4,1,1,-4,-4,1,1,-29,-29,-29,1,1,-54,-79,-79,-54,1,1,-204,
%T -254,-279,-254,-204,1,1,-479,-679,-729,-729,-679,-479,1,1,-1504,
%U -1979,-2179,-2204,-2179,-1979,-1504,1,1,-3904,-5404,-5879,-6054,-6054,-5879,-5404,-3904,1
%N Triangle T(n,k) = A015440(k) - A015440(n) + A015440(n-k), read by rows.
%C 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 G. C. Greubel, <a href="/A176263/b176263.txt">Rows n = 0..100 of triangle, flattened</a>
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, -4, 1;
%e 1, -4, -4, 1;
%e 1, -29, -29, -29, 1;
%e 1, -54, -79, -79, -54, 1;
%e 1, -204, -254, -279, -254, -204, 1;
%e 1, -479, -679, -729, -729, -679, -479, 1;
%e 1, -1504, -1979, -2179, -2204, -2179, -1979, -1504, 1;
%e 1, -3904, -5404, -5879, -6054, -6054, -5879, -5404, -3904, 1;
%p A176263 := proc(n,k)
%p A015440(k)-A015440(n)+A015440(n-k) ;
%p end proc; # _R. J. Mathar_, May 03 2013
%t (* Set of sequences q=0..10 *)
%t f[n_, q_]:= f[n, q] = If[n<2, n, f[n-1, q] + q*f[n-2, q]];
%t T[n_, k_, q_]:= f[k+1, q] + f[n-k+1, q] - f[n+1, q];
%t Table[Flatten[Table[T[n, k, q], {n,0,10}, {k,0,n}], {q,0,10}]
%t (* Second program *)
%t 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 (PARI) A015440(n) = sum(j=0,(n+1)\2, 5^j*binomial(n-j,j));
%o T(n,k) = A015440(k) - A015440(n) + A015440(n-k); \\ _G. C. Greubel_, Nov 24 2019
%o (Magma) A015440:= func< n | &+[5^j*Binomial(n-j,j): j in [0..Floor(n/2)]] >;
%o [A015440(k) - A015440(n) + A015440(n-k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 24 2019
%o (Sage)
%o def A015440(n): return sum(5^j*binomial(n-j,j) for j in (0..floor(n/2)))
%o [[A015440(k) - A015440(n) + A015440(n-k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 24 2019
%K sign,tabl,easy
%O 0,5
%A _Roger L. Bagula_, Apr 13 2010
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