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Search: id:A027307
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| A027307 |
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Number of paths from (0,0) to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1). |
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46
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| 1, 2, 10, 66, 498, 4066, 34970, 312066, 2862562, 26824386, 255680170, 2471150402, 24161357010, 238552980386, 2375085745978, 23818652359682, 240382621607874, 2439561132029314, 24881261270812490, 254892699352950850
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Equals row sums of triangle A104978 which has g.f. F(x,y) that satisfies: F = 1 + x*F^2 + x*y*F^3. - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 30 2005
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REFERENCES
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Problem 10658, American Math. Monthly, 107, 2000, 368-370.
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FORMULA
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G.f.: (2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3. a(n)=(Sum_{i=0..n-1} 2^(i+1)*binomial(2*n, i)*binomial(n, i+1))/n, n>0.
a(n) = Sum_{k=0..n} C(2*n+k, n+2*k)*C(n+2*k, k)/(n+k+1). - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 30 2005
Given g.f. A(x), y=A(x)x satisfies 0=f(x, y) where f(x, y)=x(x-y)+(x+y)y^2 . - Michael Somos May 23 2005 */
Series reversion of x(Sum_{k>=0} a(k)x^k) is x(Sum_{k>=0} A085403(k)x^k).
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PROGRAM
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(PARI) a(n)=if(n<1, n==0, sum(i=0, n-1, 2^(i+1)*binomial(2*n, i)*binomial(n, i+1))/n)
(PARI) a(n)=sum(k=0, n, binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1)) (Hanna)
(PARI) a(n)=sum(k=0, n, binomial(n, k)*binomial(2*n+k+1, n)/(2*n+k+1) ) /* Michael Somos May 23 2005 */
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CROSSREFS
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a(n)=2*A034015(n-1), n>0.
Cf. A104978.
Sequence in context: A078531 A130721 A064170 this_sequence A060206 A108205 A108397
Adjacent sequences: A027304 A027305 A027306 this_sequence A027308 A027309 A027310
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu)
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