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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 never go below the line y=x.
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4
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1, 2, 12, 88, 720, 6304, 57792, 547712, 5323008, 52761088, 531311616, 5420488704, 55905767424, 581954543616, 6106210615296, 64513688174592, 685741070942208, 7328106153115648, 78684992821788672, 848487859401261056
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Series reversion of x(1-4x)/(1-2x). - Paul Barry (pbarry(AT)wit.ie), May 19 2005
The Hankel transform of this sequence is 8^C(n+1,2)= [1,8,512,262144,...] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 08 2007
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REFERENCES
| Ira M. Gessel, A factorization for formal Laurent series and lattice path enumeration, J. Combin. Theory Ser. A 28 (1980), 321-337.
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, Jan 28, 2001. (unpublished)
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LINKS
| Robert A. Sulanke, Counting Lattice Paths by Narayana Polynomials, Electronic J. Combinatorics , Vol. 7, R40, 2000.
David Callan, A uniformly distributed statistic on a class of lattice paths, Electronic J. Combinatorics, Vol. 11(1), R82, 2004.
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FORMULA
| [1+2x-sqrt(4x^2-12x+1)]/8x
a(n)=sum{k=0..n, C(n+1, k)C(2n-k, n)(-1)^k*4^(n-k)*2^k}/(n+1); a(n)=sum{k=0..n, (1/n)*C(n, k)*C(n, k+1)*4^k*2^(n-k)}; a(n)=sum{k=0..n, N(n, k)*4^k*2^(n-k)}, N(n, k) Narayana numbers (A001263). - Paul Barry (pbarry(AT)wit.ie), May 19 2005
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MAPLE
| 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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CROSSREFS
| A006318, A001003.
a(n) = 2^(n-1)*A001003(n-1).
Cf. A054726.
a(n) = 2^n*A001003(n).
Sequence in context: A055531 A181345 A193125 * A192621 A143923 A079858
Adjacent sequences: A059432 A059433 A059434 * A059436 A059437 A059438
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KEYWORD
| nonn,easy
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AUTHOR
| Wen-jin Woan (wwoan(AT)fac.howard.edu), Feb 01 2001
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 01 2001
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