A084614 Triangle, read by rows, where the n-th row lists the (2*n+1) coefficients of (1 + x - 3*x^2)^n.
1, 1, 1, -3, 1, 2, -5, -6, 9, 1, 3, -6, -17, 18, 27, -27, 1, 4, -6, -32, 19, 96, -54, -108, 81, 1, 5, -5, -50, 5, 211, -15, -450, 135, 405, -243, 1, 6, -3, -70, -30, 366, 181, -1098, -270, 1890, -243, -1458, 729, 1, 7, 0, -91, -91, 546, 637, -2015, -1911, 4914, 2457, -7371, 0, 5103, -2187, 1, 8, 4, -112, -182, 728, 1456
Offset: 0
Examples
Rows: 1; 1, 1, -3; 1, 2, -5, -6, 9; 1, 3, -6, -17, 18, 27, -27; 1, 4, -6, -32, 19, 96, -54, -108, 81; 1, 5, -5, -50, 5, 211, -15, -450, 135, 405, -243; 1, 6, -3, -70, -30, 366, 181, -1098, -270, 1890, -243, -1458, 729;
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Programs
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Magma
A084614:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-3)^j: j in [0..k]]) >; [A084614(n,k): k in [0..2*n], n in [0..15]]; // G. C. Greubel, Mar 25 2023
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Mathematica
With[{eq= (1+x-3*x^2)}, Flatten[Table[CoefficientList[Expand[eq^n], x], {n,0,13}]]] (* G. C. Greubel, Mar 02 2017 *) -
PARI
for(n=0,12, for(k=0,2*n,t=polcoeff((1+x-3*x^2)^n,k,x); print1(t",")); print(" ")) -
SageMath
def A084614(n,k): return ( (1+x-3*x^2)^n ).series(x, 30).list()[k] flatten([[A084614(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)*(-3)^j, for 0 <= k <= 2*n.
T(n, 2*n) = (-3)^n.
T(n, 2*n-1) = (-1)^(n-1)*A027471(n+1), n >= 1.
Sum_{k=0..2*n} T(n, k) = (-1)^n.
Sum_{k=0..2*n} (-1)^k*T(n, k) = (-3)^n. (End)