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A180002
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Place a(n) red and b(n) blue balls in an urn; draw 5 balls without replacement; Probability(5 red balls) = Probability(3 red and 2 blue balls).
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3
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4, 8, 24, 43, 179, 783, 1504, 6668, 29604, 56983, 253079, 1124043, 2163724, 9610208, 42683904, 82164403, 364934699, 1620864183, 3120083464, 13857908228, 61550154924, 118481007103, 526235577839, 2337285022803, 4499158186324, 19983094049528
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OFFSET
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1,1
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COMMENTS
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This is equivalent to the Pell equation A(n)^2 - 10*B(n)^2 = -9 with a(n) = (A(n)+7)/2, b(n) = (B(n)+1)/2, and the 3 fundamental solutions (1,1), (9,3), (41,13), and the solution (19,6) for the unit form.
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LINKS
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FORMULA
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G.f.: x*(4 +4*x +16*x^2 -133*x^3 -16*x^4 -4*x^5 +3*x^6)/((1-x)*(1 -38*x^3 +x^6)).
a(n+9) = 39*a(n+6) - 39*a(n+3) + a(n).
Let r = sqrt(10) then:
a(3*n+1) = (14 + (1+r)*(19+6*r)^n + (1-r)*(19-6*r)^n)/4.
a(3*n+2) = (14 + 3*(3+r)*(19+6*r)^n + 3*(3-r)*(19-6*r)^n)/4.
a(3*n+3) = (14 + (41+13*r)*(19+6*r)^n + (41-13*r)*(19-6*r)^n)/4.
a(n) = a(n-1) + 38*a(n-3) - 38*a(n-4) - a(n-6) + a(n-7). - G. C. Greubel, Mar 20 2019
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EXAMPLE
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For n=3: a(3)=24 and b(3)=7 since binomial(24,5) = binomial(24,3)*binomial(7,2) = 42504.
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MATHEMATICA
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Rest[CoefficientList[Series[x*(4+4*x+16*x^2-133*x^3-16*x^4-4*x^5 +3*x^6 )/((1-x)*(1-38*x^3+x^6)), {x, 0, 30}], x]] (* G. C. Greubel, Mar 20 2019 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec(x*(4+4*x+16*x^2-133*x^3-16*x^4-4*x^5+3*x^6) /((1-x)*(1-38*x^3+x^6))) \\ G. C. Greubel, Mar 20 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x*(4+4*x+ 16*x^2-133*x^3-16*x^4-4*x^5+3*x^6)/((1-x)*(1-38*x^3+x^6)) )); // G. C. Greubel, Mar 20 2019
(Sage) a=(x*(4+4*x+16*x^2-133*x^3-16*x^4-4*x^5+3*x^6)/((1-x)*(1-38*x^3 +x^6))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Mar 20 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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