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A110292
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Riordan array (1-u, u) where u=(-1 + sqrt(1+8*x))/4.
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2
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1, -1, 1, 2, -3, 1, -8, 12, -5, 1, 40, -60, 26, -7, 1, -224, 336, -148, 44, -9, 1, 1344, -2016, 896, -280, 66, -11, 1, -8448, 12672, -5664, 1824, -464, 92, -13, 1, 54912, -82368, 36960, -12144, 3240, -708, 122, -15, 1, -366080, 549120, -247104, 82368, -22704, 5280, -1020, 156, -17, 1
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OFFSET
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0,4
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COMMENTS
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Inverse of Riordan array (1/(1-x), x*(1+2*x)), A110291.
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LINKS
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FORMULA
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T(n, 0) = (-1)^n * 2^(n-1) * Catalan(n-1) + (3/2)*[n=0].
T(n, n) = 1.
T(n, n-1) = 1-2*n.
T(n, 1) = (-1)^(n-1) * A181282(n-1), n >= 1.
Sum_{k=0..n} T(n, k) = A000007(n). (End)
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EXAMPLE
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Triangle begins as:
1;
-1, 1;
2, -3, 1;
-8, 12, -5, 1;
40, -60, 26, -7, 1;
-224, 336, -148, 44, -9, 1;
1344, -2016, 896, -280, 66, -11, 1;
-8448, 12672, -5664, 1824, -464, 92, -13, 1;
54912, -82368, 36960, -12144, 3240, -708, 122, -15, 1;
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MATHEMATICA
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F[k_]:= CoefficientList[Series[(5-Sqrt[1+8*x])*(-1+Sqrt[1+8*x])^k/4^(k +1), {x, 0, 20}], x];
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PROG
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(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
F:= func< k | Coefficients(R!( (5-Sqrt(1+8*x))*(-1+Sqrt(1+8*x) )^k/4^(k+1) )) >;
A110292:= func< n, k | F(k)[n-k+1] >;
(SageMath)
def p(k, x): return (5-sqrt(1+8*x))*(-1+sqrt(1+8*x))^k/4^(k+1)
def A110292(n, k): return ( p(k, x) ).series(x, 30).list()[n]
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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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