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A370343
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Expansion of g.f. A(x) satisfying 1 - Sum_{n>=1} (x^n - 3*A(x))^n = Product_{k>=1} (1 - x^(2*k)) * (1 + x^k - 3*A(x))^2 / (1 + x^(2*k) - 3*A(x))^2.
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3
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1, 5, 37, 351, 3762, 43144, 517588, 6417679, 81600076, 1058200070, 13942331746, 186108392724, 2511500635234, 34207501326618, 469639964952646, 6492479733442293, 90300079239417429, 1262676666129274167, 17740565924744385702, 250321808661666815563, 3545703074869199502562, 50398890311755064227653
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) B(x) = Product_{k>=1} (1 - x^(2*k)) * (1 + x^k - 3*A(x))^2 / (1 + x^(2*k) - 3*A(x))^2.
(2) B(x) = 1 - Sum_{n>=1} (x^n - 3*A(x))^n.
(3) B(x) = 2 - Sum_{n>=0} x^(n^2) / (1 + 3*x^n*A(x))^(n+1).
a(n) ~ c * d^n / n^(3/2), where d = 15.253891104686074069422866238429426... and c = 0.0472351025715282587234173054030... - Vaclav Kotesovec, Feb 27 2024
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EXAMPLE
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G.f.: A(x) = x + 5*x^2 + 37*x^3 + 351*x^4 + 3762*x^5 + 43144*x^6 + 517588*x^7 + 6417679*x^8 + 81600076*x^9 + 1058200070*x^10 + 13942331746*x^11 + 186108392724*x^12 + ...
By definition, A = A(x) allows for the following expressions to equal
B(x) = 1 - (x - 3*A) - (x^2 - 3*A)^2 - (x^3 - 3*A)^3 - (x^4 - 3*A)^4 - (x^5 - 3*A)^5 + ...
B(x) = (1 - x^2)*(1 + x - 3*A)^2/(1 + x^2 - 3*A)^2 * (1 - x^4)*(1 + x^2 - 3*A)^2/(1 + x^4 - 3*A)^2 * (1 - x^6)*(1 + x^3 - 3*A)^2/(1 + x^6 - 3*A)^2 * (1 - x^8)*(1 + x^4 - 3*A)^2/(1 + x^8 - 3*A)^2 * ...
where B(x) begins
B(x) = 1 + 2*x + 6*x^2 + 54*x^3 + 515*x^4 + 5478*x^5 + 62625*x^6 + 750666*x^7 + 9306054*x^8 + 118323728*x^9 + 1534492911*x^10 + 20218911516*x^11 + 269908901130*x^12 + ...
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PROG
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(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( ( 1 - sum(m=1, #A, (x^m - 3*Ser(A))^m ) - prod(m=1, #A, (1 - x^(2*m))*(1 + x^m - 3*Ser(A))^2/(1 + x^(2*m) - 3*Ser(A))^2 ) )/3, #A-1)); H=A; A[n+1]}
for(n=1, 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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