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A100218
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Riordan array ((1-2*x)/(1-x), (1-x)).
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8
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1, -1, 1, -1, -2, 1, -1, 0, -3, 1, -1, 0, 2, -4, 1, -1, 0, 0, 5, -5, 1, -1, 0, 0, -2, 9, -6, 1, -1, 0, 0, 0, -7, 14, -7, 1, -1, 0, 0, 0, 2, -16, 20, -8, 1, -1, 0, 0, 0, 0, 9, -30, 27, -9, 1, -1, 0, 0, 0, 0, -2, 25, -50, 35, -10, 1, -1, 0, 0, 0, 0, 0, -11, 55, -77, 44, -11, 1, -1, 0, 0, 0, 0, 0, 2, -36, 105, -112, 54, -12, 1
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OFFSET
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0,5
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LINKS
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FORMULA
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Sum_{k=0..n} T(n, k) = A100219(n) (row sums).
Number triangle T(n, k) = (-1)^(n-k)*(binomial(k, n-k) + binomial(k-1, n-k-1)), with T(0, 0) = 1. - Paul Barry, Nov 09 2004
T(n,k) = T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k-1) + T(n-3,k-1), T(0,0)=1, T(1,0)=-1, T(1,1)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 09 2014
T(2*n-1, n) = A157142(n-1), n >= 1.
T(3*n, n) = T(4*n, n) = A000007(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A355021(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A098601(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = -1 + 2*A077961(n) + A077961(n-2). (End)
This Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = x*(1 - x) and hence belongs to the hitting time subgroup of the Riordan group (see Peart and Woan for properties of this subgroup).
T(n,k) = [x^(n-k)] (1/c(x))^n, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. In general the (n, k)-th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)
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EXAMPLE
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Triangle begins as:
1;
-1, 1;
-1, -2, 1;
-1, 0, -3, 1;
-1, 0, 2, -4, 1;
-1, 0, 0, 5, -5, 1;
-1, 0, 0, -2, 9, -6, 1;
-1, 0, 0, 0, -7, 14, -7, 1;
-1, 0, 0, 0, 2, -16, 20, -8, 1;
-1, 0, 0, 0, 0, 9, -30, 27, -9, 1;
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MATHEMATICA
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T[0, 0]:= 1; T[1, 1]:= 1; T[1, 0]:= -1; T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, T[n- 1, k] +T[n-1, k-1] -2*T[n-2, k-1] +T[n-3, k-1]]; Table[T[n, k], {n, 0, 14}, {k, 0, n} ]//Flatten (* G. C. Greubel, Mar 13 2017 *)
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PROG
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(Magma)
A100218:= func< n, k | n eq 0 select 1 else (-1)^(n+k)*(Binomial(k, n-k) + Binomial(k-1, n-k-1)) >;
(SageMath)
def A100218(n, k): return 1 if n==0 else (-1)^(n+k)*(binomial(k, n-k) + binomial(k-1, n-k-1))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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