This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A084610 #23 Jul 17 2024 16:56:17
%S A084610 1,1,1,-1,1,2,-1,-2,1,1,3,0,-5,0,3,-1,1,4,2,-8,-5,8,2,-4,1,1,5,5,-10,
%T A084610 -15,11,15,-10,-5,5,-1,1,6,9,-10,-30,6,41,-6,-30,10,9,-6,1,1,7,14,-7,
%U A084610 -49,-14,77,29,-77,-14,49,-7,-14,7,-1,1,8,20,0,-70,-56,112,120,-125,-120,112,56,-70,0,20,-8,1
%N A084610 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+x-x^2)^n.
%H A084610 Paul D. Hanna, <a href="/A084610/b084610.txt">Rows n=0..34 of triangle, flattened</a>
%H A084610 Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, <a href="https://arxiv.org/abs/2407.04409">The Fibonacci-Fubini and Lucas-Fubini numbers</a>, arXiv:2407.04409 [math.CO], 2024. See p. 13.
%F A084610 G.f.: G(0)/2 , where G(k)= 1 + 1/( 1 - (1+x-x^2)*x^(2*k+1)/((1+x-x^2)*x^(2*k+1) + 1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 06 2013
%F A084610 From _G. C. Greubel_, Mar 26 2023: (Start)
%F A084610 T(n, k) = Sum_{j=0..k} binomial(n, k-j)*binomial(k-j, j)*(-1)^j.
%F A084610 T(n, 2*n) = (-1)^n.
%F A084610 T(n, 2*n-1) = (-1)^(n-1)*n, n >= 1.
%F A084610 Sum_{k=0..2*n} T(n, k) = 1.
%F A084610 Sum_{k=0..2*n} (-1)^k*T(n, k) = (-1)^n.
%F A084610 Sum_{k=0..n} T(n-k, k) = floor((n+2)/2).
%F A084610 Sum_{k=0..n} (-1)^k*T(n-k, k) = (-1)^n*A057597(n+2). (End)
%e A084610 Rows:
%e A084610 1;
%e A084610 1, 1, -1;
%e A084610 1, 2, -1, -2, 1;
%e A084610 1, 3, 0, -5, 0, 3, -1;
%e A084610 1, 4, 2, -8, -5, 8, 2, -4, 1;
%e A084610 1, 5, 5, -10, -15, 11, 15, -10, -5, 5, -1;
%e A084610 1, 6, 9, -10, -30, 6, 41, -6, -30, 10, 9, -6, 1;
%e A084610 1, 7, 14, -7, -49, -14, 77, 29, -77, -14, 49, -7, -14, 7, -1;
%t A084610 T[n_, k_]:= Sum[Binomial[n,k-j]*Binomial[k-j,j]*(-1)^j, {j,0,k}];
%t A084610 Table[T[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* _G. C. Greubel_, Mar 26 2023 *)
%o A084610 (PARI) for(n=0,12, for(k=0,2*n,t=polcoeff((1+x-x^2)^n,k,x); print1(t",")); print(" "))
%o A084610 (Magma)
%o A084610 A084610:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-1)^j: j in [0..k]]) >;
%o A084610 [A084610(n,k): k in [0..2*n], n in [0..13]]; // _G. C. Greubel_, Mar 26 2023
%o A084610 (SageMath)
%o A084610 def A084610(n,k): return sum(binomial(n,k-j)*binomial(k-j,j)*(-1)^j for j in range(k+1))
%o A084610 flatten([[A084610(n,k) for k in range(2*n+1)] for n in range(14)]) # _G. C. Greubel_, Mar 26 2023
%Y A084610 Cf. A002426, A057597, A084600 - A084609, A084611 - A084615.
%K A084610 sign,tabf
%O A084610 0,6
%A A084610 _Paul D. Hanna_, Jun 01 2003