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1, 3, 51, 1119, 29103, 859143, 28091463, 1002057591, 38606468343, 1595167432599, 70315835952471, 3293268346004439, 163337193581191575, 8554718468806548951, 471976737725208306327, 27369722655919760451159, 1664858070989667129693975, 106029602841882346657155543
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OFFSET
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0,2
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LINKS
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FORMULA
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Let t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).
O.g.f.: A(x) = ( Sum_{k >= 0} t(k+4)/t(4)*x^k )/( Sum_{k >= 0} t(k+3)/t(3)*x^k ).
A(x)/(1 - 11*x*A(x)) = Sum_{k >= 0} t(k+4)/t(4)*x^k.
A(x) = 1/(1 + 11*x - 14*x/(1 + 14*x -17*x/(1 + 17*x - 20*x/(1 + 20*x - ... )))) (continued fraction).
A(x) satisfies the Riccati differential equation 3*x^2*d/dx(A(x)) + 11*x*A(x)^2 - (1 + 8*x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 3*x/(1 - 14*x/(1 - 6*x/(1 - 17*x/(1 - 9*x/(1 - 20*x/(1 - 12*x/(1 - 23*x/(1 - ...))))))))), a continued fraction of Stieltjes type.
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MAPLE
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n := 4: seq(coeff(series( hypergeom([n+2/3, 1], [], 3*x)/hypergeom([n-1/3, 1], [], 3*x ), x, 21), x, k), k = 0..20);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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