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A135052
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Expansion of (1-2x-sqrt(1-4x+8x^3-4x^4))/(2x^2(1-x)).
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1
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1, 1, 3, 7, 19, 51, 143, 407, 1183, 3487, 10415, 31439, 95791, 294191, 909823, 2830943, 8856255, 27839167, 87888767, 278545663, 885903743, 2826612095, 9045147391, 29022168063, 93350430975, 300949170431, 972271227647
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Sequence is the binomial transform of the aerated large Schroeder numbers A006318. Hankel transform is A060656(n+1).
a(n)=number of lattice paths, never going below the x-axis, from (0,0) to (n,0) consisting of up steps U = (1,1), down steps D = (1,-1) and horizontal steps H(k) = (k,0) for every positive integer k. For instance, for n=3, we have the 7 paths: H(1)H(1)H(1), H(1)H(2), H(2)H(1), H(3), H(1)UD, UDH(1), UH(1)D. - Emanuele Munarini, Mar 14 2011.
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FORMULA
| a(n)=sum{k=0..n, sum{j=0..k/2, C(k/2+j, 2j)*C(j)*(1+(-1)^k)/2}}, where C(n)=A000108(n).
G.f.: 1/(1-x-2x^2/(1-x-x^2/(1-x-2x^2/(1-x-x^2/(1-x-2x^2.... (continued fraction); [From Paul Barry (pbarry(AT)wit.ie), Jan 02 2009]
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CROSSREFS
| Sequence in context: A002426 A011769 A087432 * A198305 A146597 A115254
Adjacent sequences: A135049 A135050 A135051 * A135053 A135054 A135055
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Nov 15 2007
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