A084612 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x - 2*x^2)^n.
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, 5, 0, -30, -15, 81, 30, -120, 0, 80, -32, 1, 6, 3, -40, -45, 126, 141, -252, -180, 320, 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
Offset: 0
Examples
Triangle begins: 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, 5, 0, -30, -15, 81, 30, -120, 0, 80, -32; 1, 6, 3, -40, -45, 126, 141, -252, -180, 320, 48, -192, 64;
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..1023
Programs
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Magma
A084612:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-2)^j: j in [0..k]]) >; [A084612(n,k): k in [0..2*n], n in [0..13]]; // G. C. Greubel, Mar 25 2023
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Mathematica
T[n_, k_]:= Sum[Binomial[n,k-j]*Binomial[k-j,j]*(-2)^j, {j,0,k}]; Table[T[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* G. C. Greubel, Mar 25 2023 *) -
PARI
{T(n,k)=polcoeff((1+x-2*x^2)^n, k)} for(n=0,10,for(k=0,2*n,print1(T(n,k),", "));print("")) -
SageMath
def A084612(n,k): return sum(binomial(n,k-j)*binomial(k-j,j)*(-2)^j for j in range(k+1)) flatten([[A084612(n,k) for k in range(2*n+1)] for n in range(13)]) # G. C. Greubel, Mar 25 2023
Formula
From G. C. Greubel, Mar 25 2023: (Start)
T(n, k) = Sum_{j=0..k} binomial(n,k-j)*binomial(k-j,j)*(-2)^j, for 0 <= k <= 2*n.
T(n, 2*n) = (-2)^n.
T(n, 2*n-1) = (-1)^(n-1)*A001787(n), n >= 1.
Sum_{k=0..2*n} T(n, k) = A000007(n).
Sum_{k=0..2*n} (-1)^k*T(n, k) = (-2)^n. (End)