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A108442
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Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and having only u steps among the steps leading to the first d step.
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2
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1, 1, 3, 15, 97, 721, 5827, 49759, 441729, 4035937, 37702723, 358474735, 3457592161, 33748593841, 332730216579, 3308635650495, 33145196426753, 334193815799233, 3388807714823043, 34537227997917391, 353578650475659617
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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REFERENCES
| Problem 10658, American Math. Monthly, 107, 2000, 368-370.
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FORMULA
| G.f.=1/(1-zA), where A=1+zA^2+zA^3=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
a(n)=sum(k=1..n, (k*sum(i=0..n-k, binomial(2*n-k,i)*binomial(3*n-2*k-i-1,2*n-k-1)))/(2*n-k)), n>0, a(0)=1. [From Vladimir Kruchinin, Oct 23 2011]
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EXAMPLE
| a(2)=3 because we have udud, udUdd and uudd.
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MAPLE
| A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: gser:=series(1/(1-z*A), z=0, 30): 1, seq(coeff(gser, z^n), n=1..25);
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PROG
| (Maxima)
a(n):=if n=0 then 1 else sum((k*sum(binomial(2*n-k, i)*binomial(3*n-2*k-i-1, 2*n-k-1), i, 0, n-k))/(2*n-k), k, 1, n); [From Vladimir Kruchinin, Oct 23 2011]
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CROSSREFS
| Column 0 of A108441.
Cf. A027307, A108441.
Sequence in context: A112913 A109283 A079689 * A060148 A143435 A132437
Adjacent sequences: A108439 A108440 A108441 * A108443 A108444 A108445
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 08 2005
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