A104402 Matrix inverse of triangle A091491, read by rows.
1, -1, 1, 1, -2, 1, 0, 2, -3, 1, 0, -1, 4, -4, 1, 0, 0, -3, 7, -5, 1, 0, 0, 1, -7, 11, -6, 1, 0, 0, 0, 4, -14, 16, -7, 1, 0, 0, 0, -1, 11, -25, 22, -8, 1, 0, 0, 0, 0, -5, 25, -41, 29, -9, 1, 0, 0, 0, 0, 1, -16, 50, -63, 37, -10, 1, 0, 0, 0, 0, 0, 6, -41, 91, -92, 46, -11, 1, 0, 0, 0, 0, 0, -1, 22, -91, 154, -129, 56, -12, 1
Offset: 0
Examples
Triangle begins as: 1; -1, 1; 1, -2, 1; 0, 2, -3, 1; 0, -1, 4, -4, 1; 0, 0, -3, 7, -5, 1; 0, 0, 1, -7, 11, -6, 1; 0, 0, 0, 4, -14, 16, -7, 1; 0, 0, 0, -1, 11, -25, 22, -8, 1;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Mathematica
Table[(-1)^(n-k)*(Binomial[k, n-k] + Binomial[k+1, n-k-1]), {n,0,12}, {k,0,n}] //Flatten (* G. C. Greubel, Apr 30 2021 *) -
PARI
T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1-X+X^2)/(1-X*Y*(1-X)),n,x),k,y)
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PARI
T(n,k)=if(n
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PARI
T(n,k)=(-1)^(n-k)*(binomial(k,n-k)+binomial(k+1,n-k-1))
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Sage
def A104402(n,k): return (-1)^(n+k)*(binomial(k,n-k) + binomial(k+1,n-k-1)) flatten([[A104402(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 30 2021
Formula
G.f.: (1-x+x^2)/(1-x*y*(1-x)).
T(n, k) = T(n-1, k-1) - T(n-2, k-1) for k>0 with T(0, 0)=1, T(1, 0)=-1, T(2, 0)=1, T(n, 0)=0 (n>2).
T(n, k) = (-1)^(n-k)*(C(k, n-k) + C(k+1, n-k-1)).
From Philippe Deléham, Nov 04 2009: (Start)
Sum_{k=0..n} T(n,k) = 0^n.
Sum_{k=0..n} abs(T(n, k)) = 2*Fibonacci(n+1) - [n=0].
Sum_{k=0..n} ( T(n,k) )^2 = 2*A003440(n) - [n=0]. (End)
Comments