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A176264 Triangle T(n,k) = A015442(k) - A015442(n) + A015442(n-k) read by rows.

%I A176264 #12 Jun 29 2025 18:28:31
%S A176264 1,1,1,1,-6,1,1,-6,-6,1,1,-55,-55,-55,1,1,-104,-153,-153,-104,1,1,
%T A176264 -496,-594,-643,-594,-496,1,1,-1231,-1721,-1819,-1819,-1721,-1231,1,1,
%U A176264 -4710,-5935,-6425,-6474,-6425,-5935,-4710,1,1,-13334,-18038,-19263,-19704,-19704,-19263,-18038,-13334,1
%N A176264 Triangle T(n,k) = A015442(k) - A015442(n) + A015442(n-k) read by rows.
%C A176264 Row sums are s(n) = {1, 2, -4, -10, -163, -512, -2821, -9540, -40612, -140676, -537533, ...} where s(n) = 3*s(n-1) +11*s(n-2) -27*s(n-3) -35*s(n-4) +49*s(n-5) with g.f. (1-x-21*x^2+7*x^3)/((1-x)*(1-x-7*x^2)^2).
%H A176264 G. C. Greubel, <a href="/A176264/b176264.txt">Rows n = 0..100 of triangle, flattened</a>
%F A176264 T(n,k) = T(n,n-k).
%e A176264 Triangle begins as:
%e A176264   1;
%e A176264   1,      1;
%e A176264   1,     -6,      1;
%e A176264   1,     -6,     -6,      1;
%e A176264   1,    -55,    -55,    -55,      1;
%e A176264   1,   -104,   -153,   -153,   -104,      1;
%e A176264   1,   -496,   -594,   -643,   -594,   -496,      1;
%e A176264   1,  -1231,  -1721,  -1819,  -1819,  -1721,  -1231,      1;
%e A176264   1,  -4710,  -5935,  -6425,  -6474,  -6425,  -5935,  -4710,      1;
%e A176264   1, -13334, -18038, -19263, -19704, -19704, -19263, -18038, -13334,      1;
%e A176264   1, -46311, -59639, -64343, -65519, -65911, -65519, -64343, -59639, -46311, 1;
%t A176264 (* Set of sequences q=0..10. This sequence is q=7. *)
%t A176264 f[n_, q_]:= f[n, q] = If[n<2, n, f[n-1, q] + q*f[n-2, q]];
%t A176264 T[n_, k_, q_]:= f[k+1, q] + f[n-k+1, q] - f[n+1, q];
%t A176264 Table[Flatten[Table[T[n, k, q], {n,0,10}, {k,0,n}]], {q,0,10}]
%t A176264 (* Second program *)
%t A176264 A015442[n_]:= Sum[7^j*Binomial[n-j, j], {j,0,(n+1)/2}]; T[n_, k_]:= T[n, k]= A015442[k] +A015442[n-k] -A015442[n]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 24 2019 *)
%o A176264 (PARI) A015442(n) = sum(j=0,(n+1)\2, 7^j*binomial(n-j,j));
%o A176264 T(n,k) = A015442(k) - A015442(n) + A015442(n-k); \\ _G. C. Greubel_, Nov 24 2019
%o A176264 (Magma) A015442:= func< n | &+[7^j*Binomial(n-j,j): j in [0..Floor(n/2)]] >;
%o A176264 [A015442(k) - A015442(n) + A015442(n-k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 24 2019
%o A176264 (Sage)
%o A176264 def A015442(n): return sum(7^j*binomial(n-j,j) for j in (0..floor(n/2)))
%o A176264 [[A015442(k) - A015442(n) + A015442(n-k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 24 2019
%K A176264 sign,tabl,easy
%O A176264 0,5
%A A176264 _Roger L. Bagula_, Apr 13 2010