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A059435
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Number of lattice paths in plane starting at (0,0) and ending at (n,n) with steps from {(i,j): i+j > 0, i, j >= 0} that never go below the line y = x.
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6
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1, 2, 12, 88, 720, 6304, 57792, 547712, 5323008, 52761088, 531311616, 5420488704, 55905767424, 581954543616, 6106210615296, 64513688174592, 685741070942208, 7328106153115648, 78684992821788672, 848487859401261056
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OFFSET
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0,2
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COMMENTS
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Series reversion of x(1-4x)/(1-2x). - Paul Barry, May 19 2005
The Hankel transform of this sequence is 8^C(n+1,2) = [1, 8, 512, 262144, ...]. - Philippe Deléham, Nov 08 2007
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REFERENCES
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W.-J. Woan, A bijective proof by induction that the n-th term of this sequence is 2^(n-1) times of the n-th term of the big Schroeder number, 2001 (unpublished).
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LINKS
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FORMULA
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G.f.: (1 + 2*x - sqrt(4*x^2 - 12*x + 1))/(8*x).
a(n) = (1/(n + 1)) * Sum_{k=0..n} C(n+1, k) * C(2*n-k, n)(-1)^k * 4^(n-k) * 2^k;
a(n) = Sum_{k=0..n} (1/n) * C(n, k) * C(n, k+1) * 4^k * 2^(n-k);
a(n) = Sum_{k=1..n} N(n, k)*2^(n+k-1)}, for n >= 1, where N(n, k) are the Narayana numbers (A001263). [Corrected by Alejandro H. Morales, May 14 2015]
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Recurrence: (n+1)*a(n) = 6*(2*n-1)*a(n-1) - 4*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 11 2012
a(n) ~ sqrt(4+3*sqrt(2))*(6+4*sqrt(2))^n/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 11 2012
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MAPLE
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gf := (1+2*x-sqrt(4*x^2-12*x+1))/(8*x): s := series(gf, x, 100): for i from 0 to 50 do printf(`%d, `, coeff(s, x, i)) od:
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MATHEMATICA
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Table[SeriesCoefficient[(1+2*x-Sqrt[4*x^2-12*x+1])/(8*x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 11 2012 *)
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PROG
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(PARI) x='x+O('x^66); Vec((1+2*x-sqrt(4*x^2-12*x+1))/(8*x)) \\ Joerg Arndt, May 06 2013
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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