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A084612 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x - 2*x^2)^n.

%I A084612 #14 Mar 26 2023 03:50:14
%S A084612 1,1,1,-2,1,2,-3,-4,4,1,3,-3,-11,6,12,-8,1,4,-2,-20,1,40,-8,-32,16,1,
%T A084612 5,0,-30,-15,81,30,-120,0,80,-32,1,6,3,-40,-45,126,141,-252,-180,320,
%U A084612 48,-192,64,1,7,7,-49,-91,161,357,-363,-714,644,728,-784,-224,448,-128,1,8,12,-56,-154,168,700,-328,-1791,656,2800
%N A084612 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x - 2*x^2)^n.
%H A084612 Paul D. Hanna, <a href="/A084612/b084612.txt">Table of n, a(n) for n = 0..1023</a>
%F A084612 From _G. C. Greubel_, Mar 25 2023: (Start)
%F A084612 T(n, k) = Sum_{j=0..k} binomial(n,k-j)*binomial(k-j,j)*(-2)^j, for 0 <= k <= 2*n.
%F A084612 T(n, 2*n) = (-2)^n.
%F A084612 T(n, 2*n-1) = (-1)^(n-1)*A001787(n), n >= 1.
%F A084612 Sum_{k=0..2*n} T(n, k) = A000007(n).
%F A084612 Sum_{k=0..2*n} (-1)^k*T(n, k) = (-2)^n. (End)
%e A084612 Triangle begins:
%e A084612   1;
%e A084612   1, 1, -2;
%e A084612   1, 2, -3,  -4,   4;
%e A084612   1, 3, -3, -11,   6,  12,  -8;
%e A084612   1, 4, -2, -20,   1,  40,  -8,  -32,   16;
%e A084612   1, 5,  0, -30, -15,  81,  30, -120,    0,  80, -32;
%e A084612   1, 6,  3, -40, -45, 126, 141, -252, -180, 320,  48, -192, 64;
%t A084612 T[n_, k_]:= Sum[Binomial[n,k-j]*Binomial[k-j,j]*(-2)^j, {j,0,k}];
%t A084612 Table[T[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* _G. C. Greubel_, Mar 25 2023 *)
%o A084612 (PARI) {T(n,k)=polcoeff((1+x-2*x^2)^n, k)}
%o A084612 for(n=0,10,for(k=0,2*n,print1(T(n,k),", "));print(""))
%o A084612 (Magma)
%o A084612 A084612:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-2)^j: j in [0..k]]) >;
%o A084612 [A084612(n,k): k in [0..2*n], n in [0..13]]; // _G. C. Greubel_, Mar 25 2023
%o A084612 (SageMath)
%o A084612 def A084612(n,k): return sum(binomial(n,k-j)*binomial(k-j,j)*(-2)^j for j in range(k+1))
%o A084612 flatten([[A084612(n,k) for k in range(2*n+1)] for n in range(13)]) # _G. C. Greubel_, Mar 25 2023
%Y A084612 Cf. A000007, A001787, A002426, A084600 - A084611, A084613, A084614, A084615.
%K A084612 sign,tabf
%O A084612 0,4
%A A084612 _Paul D. Hanna_, Jun 01 2003