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A164948
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Fibonacci matrix read by antidiagonals. (Inverse of A136158.)
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3
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1, 1, -1, 3, -4, 1, 9, -15, 7, -1, 27, -54, 36, -10, 1, 81, -189, 162, -66, 13, -1, 243, -648, 675, -360, 105, -16, 1, 729, -2187, 2673, -1755, 675, -153, 19, -1, 2187, -7290, 10206, -7938, 3780, -1134, 210, -22, 1, 6561, -24057, 37908, -34020, 19278, -7182, 1764, -276, 25, -1, 19683, -78732, 137781, -139968, 91854, -40824, 12474, -2592, 351, -28, 1
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OFFSET
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0,4
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COMMENTS
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Triangle, read by rows, given by [1,2,0,0,0,0,0,0,0,...] DELTA [-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 02 2009
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LINKS
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FORMULA
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Sum_{k=0..floor(n/2)} T(n-k, k) = A001519(n).
T(n,k) = 3*T(n-1,k) - T(n-1,k-1) with T(0,0)=1, T(1,0)=1, T(1,1)=-1.
Row n: Expansion of (1-x)*(3-x)^(n-1), n>0. (End)
T(n, k) = (-1)^k*3^(n-k-1)*((n+2*k)/n)*binomial(n, k), for n > 0, with T(0, 0) = 1.
T(2*n, n) = 4*(-1)^n*A098399(n-1) - (1/3)*[n=0].
T(n, n-4) = 27*(-1)^n*A001296(n-3), n >= 4.
T(n, n-3) = 9*(-1)^(n-1)*A002411(n-2), n >= 3.
T(n, n-1) = (-1)^(n-1)*A016777(n-1), n >= 1.
T(n, n) = (-1)^n.
Sum_{k=0..n} (-1)^k*T(n, k) = A081294(n).
Sum_{k=0..n} (-1)^k*T(n-k, k) = A003688(n). (End)
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EXAMPLE
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As triangle:
1;
1, -1;
3, -4, 1;
9, -15, 7, -1;
27, -54, 36, -10, 1;
81, -189, 162, -66, 13, -1;
243, -648, 675, -360, 105, -16, 1;
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MATHEMATICA
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A164948[n_, k_]:= If[n==0, 1, (-1)^k*3^(n-k-1)*(n+2*k)*Binomial[n, k]/n];
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PROG
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(Magma)
A164948:= func< n, k | n eq 0 select 1 else (-1)^k*3^(n-k-1)*(n+2*k)*Binomial(n, k)/n >;
(SageMath)
def A164948(n, k): return 1 if (n==0) else (-1)^k*3^(n-k-1)*((n+2*k)/n)*binomial(n, k)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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