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