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A292814
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G.f.: A(x) satisfies: A( 6*x - 5*A(x) ) = x - 16*x^2.
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8
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1, 4, -40, 1400, -68000, 4350000, -335400000, 30085500000, -3064470000000, 348742875000000, -43821408750000000, 6025223212500000000, -899952237525000000000, 145148307600812500000000, -25148944415520000000000000, 4660143903279037500000000000, -919883216910828187500000000000, 192745988160330338593750000000000, -42734275076178265882812500000000000, 9996684603960098310695312500000000000, -2460868592009779893077500000000000000000
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OFFSET
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1,2
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COMMENTS
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Conjecture: In general, if k > 0 and g.f. A(x) satisfies A((k+2)*x - (k+1)*A(x)) = x - (k*x)^2, then a(n) ~ (-1)^n * c(k) * (k*(k+1)/log(k+1))^n * n! / n^((k-1)/(k+1) + (k-2)*log(k+1)/k), where c(k) is a constant. - Vaclav Kotesovec, Oct 09 2017
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LINKS
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FORMULA
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a(n) ~ (-1)^n * c * 20^n * n! / (n^(3/5 + log(5)/2) * (log(5))^n), where c = 0.04310841216382017737... - Vaclav Kotesovec, Oct 09 2017
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EXAMPLE
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G.f.: A(x) = x + 4*x^2 - 40*x^3 + 1400*x^4 - 68000*x^5 + 4350000*x^6 - 335400000*x^7 + 30085500000*x^8 - 3064470000000*x^9 + 348742875000000*x^10 - 43821408750000000*x^11 + 6025223212500000000*x^12 - 899952237525000000000*x^13 + 145148307600812500000000*x^14 - 25148944415520000000000000*x^15 +...
such that A( 6*x - 5*A(x) ) = x - 16*x^2.
RELATED SERIES.
Define Ai(x) such that Ai(A(x)) = x, then Ai(x) begins:
Ai(x) = x - 4*x^2 + 72*x^3 - 2520*x^4 + 123424*x^5 - 7710448*x^6 + 579543872*x^7 - 50689868256*x^8 + 5046335530880*x^9 - 562904575158208*x^10 +...
where Ai(x - 16*x^2) = 6*x - 5*A(x).
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PROG
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(PARI) {a(n) = my(A=x, V=[1, 4]); for(i=1, n, V = concat(V, 0); A=x*Ser(V); V[#V] = Vec( subst(A, x, 6*x - 5*A) )[#V]/4 ); V[n]}
for(n=1, 30, 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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