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A291614
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G.f. A(x) satisfies: A( x*A(x)^2 - x^2*A(x) ) = x^4.
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2
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1, 1, -1, 2, -5, 13, -40, 126, -409, 1363, -4617, 15896, -55444, 195480, -695636, 2495118, -9011281, 32741839, -119601339, 438968354, -1618006837, 5986803522, -22229028994, 82798248894, -309299225632, 1158483827048, -4349740078410, 16368842606820, -61727972554068, 233233590724532, -882851162632794, 3347489178793192, -12712739206990305
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OFFSET
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1,4
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LINKS
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FORMULA
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a(n) ~ (-1)^n * c * d^n / n^(3/2), where d = 3.98431348330228671611917... and c = 0.0351241010191430532... - Vaclav Kotesovec, Aug 28 2017
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EXAMPLE
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G.f.: A(x) = x + x^2 - x^3 + 2*x^4 - 5*x^5 + 13*x^6 - 40*x^7 + 126*x^8 - 409*x^9 + 1363*x^10 - 4617*x^11 + 15896*x^12 - 55444*x^13 + 195480*x^14 - 695636*x^15 + 2495118*x^16 - 9011281*x^17 + 32741839*x^18 - 119601339*x^19 + 438968354*x^20 +...
such that A( x*A(x)^2 - x^2*A(x) ) = x^4.
RELATED SERIES.
x*A(x)^2 - x^2*A(x) = x^4 - x^8 + 3*x^12 - 12*x^16 + 55*x^20 - 272*x^24 +...
Let Ai(x) denote the series reversion of A(x), then
Ai(x) = x - x^2 + 3*x^3 - 12*x^4 + 55*x^5 - 272*x^6 + 1418*x^7 - 7674*x^8 + 42703*x^9 - 242802*x^10 + 1404430*x^11 - 8238084*x^12 + 48888198*x^13 - 292986110*x^14 + 1770676784*x^15 - 10779271798*x^16 + 66038787055*x^17 - 406856161566*x^18 + 2519083078066*x^19 - 15666603475448*x^20 +...
where Ai( Ai(x)^4 ) = x^2*Ai(x) - x*Ai(x)^2.
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PROG
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(PARI) {a(n) = my(A=[1, 1, -1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = polcoeff(x^4 - subst(F, x, x*F^2 - x^2*F), #A+2) ); A[n]}
for(n=1, 40, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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