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A305599
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G.f. A(x) satisfies: 0 = Sum_{n>=1} (n*x - A(x))^n, with A(0) = 0, A'(0) = 1.
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1
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1, 1, 6, 58, 746, 11786, 219076, 4664939, 111755188, 2972722571, 86921839164, 2771315128064, 95702436112958, 3559390186217691, 141876840484813540, 6034767098100363938, 272876063329174663702, 13072133069784764308105, 661414882620924186276282, 35249032476818384801858422, 1973681124359292990559785042, 115843427877914768308868529869
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = x*[Sum_{n>=1} n * (n*x - A(x))^(n-1)] / [Sum_{n>=1} (n*x - A(x))^(n-1)].
(2) A'(x) = [Sum_{n>=1} n^2 * (n*x - A(x))^(n-1)] / [Sum_{n>=1} n * (n*x - A(x))^(n-1)].
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EXAMPLE
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G.f.: A(x) = x + x^2 + 6*x^3 + 58*x^4 + 746*x^5 + 11786*x^6 + 219076*x^7 + 4664939*x^8 + 111755188*x^9 + 2972722571*x^10 + ...
such that
0 = (x - A(x)) + (2*x - A(x))^2 + (3*x - A(x))^3 + (4*x - A(x))^4 + (5*x - A(x))^5 + (6*x - A(x))^6 + (7*x - A(x))^7 + ...
RELATED SERIES.
D(x) = Sum_{n>=1} (n*x - A(x))^(n-1) = 1 + x + 3*x^2 + 17*x^3 + 148*x^4 + 1750*x^5 + 26014*x^6 + 462851*x^7 + 9545839*x^8 + 223275067*x^9 + 5830940099*x^10 + ...
F(x) = Sum_{n>=1} n * (n*x - A(x))^(n-1) = 1 + 2*x + 10*x^2 + 84*x^3 + 987*x^4 + 14706*x^5 + 262738*x^6 + 5440004*x^7 + 127654399*x^8 + 3342353520*x^9 + ...
where x * F(x) / D(x) = A(x).
G(x) = Sum_{n>=1} n^2 * (n*x - A(x))^(n-1) = 1 + 4*x + 32*x^2 + 372*x^3 + 5529*x^4 + 98688*x^5 + 2041668*x^6 + 47866228*x^7 + 1252068173*x^8 + 36122018580*x^9 + ...
where G(x) / F(x) = A'(x), which begins
A'(x) = 1 + 2*x + 18*x^2 + 232*x^3 + 3730*x^4 + 70716*x^5 + 1533532*x^6 + 37319512*x^7 + 1005796692*x^8 + 29727225710*x^9 + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff(sum(m=1, #A, (m*x - x*Ser(A))^m), #A)); A[n]}
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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