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A110165
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Riordan array (1/sqrt(1-6x+5x^2),(1-3x-sqrt(1-6x+5x^2))/(2x)).
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6
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1, 3, 1, 11, 6, 1, 45, 30, 9, 1, 195, 144, 58, 12, 1, 873, 685, 330, 95, 15, 1, 3989, 3258, 1770, 630, 141, 18, 1, 18483, 15533, 9198, 3801, 1071, 196, 21, 1, 86515, 74280, 46928, 21672, 7210, 1680, 260, 24, 1, 408105, 356283, 236736, 119154, 44982, 12510, 2484, 333, 27, 1
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OFFSET
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0,2
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COMMENTS
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This Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = (1 - 3*x - sqrt(1 - 6*x + 5*x^2))/(2*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = 1 + 3*x + x^2. 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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LINKS
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FORMULA
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Number triangle T(n, k) = Sum_{j = 0..n} C(n, j)C(2j, j+k).
T(n,0) = 3*T(n-1,0) + 2*T(n-1,1), T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + T(n-1,k+1) for k > 0, T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 24 2014
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EXAMPLE
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Rows begin
1;
3, 1;
11, 6, 1;
45, 30, 9, 1;
195, 144, 58, 12, 1;
873, 685, 330, 95, 15, 1;
Production array begins:
3, 1;
2, 3, 1;
0, 1, 3, 1;
0, 0, 1, 3, 1;
0, 0, 0, 1, 3, 1;
0, 0, 0, 0, 1, 3, 1;
0, 0, 0, 0, 0, 1, 3, 1;
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MAPLE
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seq(seq( coeff((x^2 + 3*x + 1)^n, x, n-k), k = 0..n ), n = 0..10); # Peter Bala, Jan 09 2022
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MATHEMATICA
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(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/Sqrt[1-6#+5#^2]&, (1-3#-Sqrt[1-6#+5#^2])/(2#)&, 10] // Flatten (* Jean-François Alcover, Jul 19 2019 *)
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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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