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A211855
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G.f. satisfies: A(x) = (1+x*A(x)^3)*(1+x^2*A(x)^2)*(1+x^3*A(x)).
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3
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1, 1, 4, 19, 98, 553, 3288, 20287, 128681, 833889, 5496837, 36742204, 248454438, 1696588460, 11682677436, 81031854579, 565614332353, 3970182041035, 28006229772030, 198438070511163, 1411652452459443, 10078529348799106, 72192155099054325, 518659038159324250
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) ~ sqrt(s*(2*r*s + s^2 + 5*r^4*s^2 + 4*r^3*s^3 + 6*r^5*s^5 + 3*r^2*(1 + s^4)) / (Pi*(r + 3*s + 3*r^4*s + 6*r^3*s^2 + 10*r^2*s^3 + 15*r^5*s^4))) / (2*n^(3/2)*r^n), where r = 0.1303652752058746790368151406944165350206179676971... and s = 1.504659035764367744283558911063644754705733371817... are real roots of the system of equations (1 + r^3*s)*(1 + r^2*s^2)*(1 + r*s^3) = s, r*(2*r*s + 3*s^2 + 3*r^4*s^2 + 4*r^3*s^3 + 6*r^5*s^5 + r^2*(1 + 5*s^4)) = 1. - Vaclav Kotesovec, Nov 22 2017
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EXAMPLE
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G.f.: A(x) = 1 + x + 4*x^2 + 19*x^3 + 98*x^4 + 553*x^5 + 3288*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 9*x^2 + 46*x^3 + 250*x^4 + 1454*x^5 + 8827*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 82*x^3 + 468*x^4 + 2808*x^5 + 17431*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 128*x^3 + 765*x^4 + 4736*x^5 + 30086*x^6 +...
A(x)^5 = 1 + 5*x + 30*x^2 + 185*x^3 + 1155*x^4 + 7376*x^5 + 47970*x^6 +...
A(x)^6 = 1 + 6*x + 39*x^2 + 254*x^3 + 1653*x^4 + 10884*x^5 + 72474*x^6 +...
where A(x) = 1 + x*A(x)^3 + x^2*A(x)^2 + x^3*(A(x)+A(x)^5) + x^4*A(x)^4 + x^5*A(x)^3 + x^6*A(x)^6.
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+x*A^3)*(1+x^2*A^2)*(1+x^3*A)+x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, 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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