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A228411
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G.f.: ( (1 - sqrt(1-32*x)) / (16*x) )^(1/4).
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2
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1, 2, 26, 476, 10150, 236060, 5807076, 148581048, 3913759878, 105424703020, 2890693930124, 80413849328904, 2263896023453532, 64381391412987672, 1846729385267277960, 53367451809002583408, 1552274439636853988550, 45408989873571191613900, 1335107241077282661195900
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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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G.f. A(x) satisfies:
(1) A(x) = exp( x*A(x)^8 + Integral(A(x)^8 dx) ).
(2) A(x)^4 = 1 + 8*x*A(x)^8, thus A(x) = C(8*x)^(1/4) where C(x) is the Catalan function (A000108).
D-finite with recurrence: n*(4*n+1)*a(n) -2*(8*n-3)*(8*n-7)*a(n-1)=0. - R. J. Mathar, Oct 08 2016
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 26*x^2 + 476*x^3 + 10150*x^4 + 236060*x^5 +...
where
A(x)^4 = 1 + 8*x + 128*x^2 + 2560*x^3 + 57344*x^4 + 1376256*x^5 +...+ A000108(n)*8^n*x^n +...
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MATHEMATICA
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CoefficientList[Series[((1-Sqrt[1-32*x])/(16*x))^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 10 2013 *)
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PROG
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(PARI) /* G.f.: ( (1 - sqrt(1-32*x)) / (16*x) )^(1/4): */
{a(n)=polcoeff(( (1 - sqrt(1-32*x +x^2*O(x^n))) / (16*x) )^(1/4), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* G.f.: A(x) = C(8*x)^(1/4), C(x) is Catalan function: */
{a(n)=polcoeff((serreverse(x-8*x^2 +x^2*O(x^n))/x)^(1/4), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* G.f.: A(x) = exp( x*A(x)^8 + Integral(A(x)^8 dx) ): */
{a(n)=local(A=1+x); for(i=1, n, A=exp(x*A^8+intformal(A^8+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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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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