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A357398
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a(n) = coefficient of x^n/n!, n >= 0, in A(x) such that: 0 = Sum_{n>=1} exp(-n^2*x) * (exp(n*x) - A(x))^n.
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2
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1, 1, 3, 37, 1083, 53701, 3934443, 395502997, 51998075643, 8643190760101, 1770707733052683, 438247927304923957, 128926370847248904603, 44477192002157773868101, 17787359321176954021105323, 8164879801560415752441320917, 4264616618784184682736855291963
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OFFSET
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0,3
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COMMENTS
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All terms appear to be odd.
Conjecture: after initial term, sequence modulo 10 has period 4: [1,3,7,3] repeating.
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LINKS
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2/2! + 37*x^3/3! + 1083*x^4/4! + 53701*x^5/5! + 3934443*x^6/6! + 395502997*x^7/7! + 51998075643*x^8/8! + 8643190760101*x^9/9! + 1770707733052683*x^10/10! + ...
where
0 = exp(-x)*(exp(x) - A(x)) + exp(-4*x)*(exp(2*x) - A(x))^2 + exp(-9*x)*(exp(3*x) - A(x))^3 + exp(-16*x)*(exp(4*x) - A(x))^4 + exp(-25*x)*(exp(5*x) - A(x))^5 + ... + exp(-n^2*x) * (exp(x)^n - A(x))^n + ...
equivalently,
0 = (1 - exp(-x)*A(x)) + (1 - exp(-2*x)*A(x))^2 + (1 - exp(-3*x)*A(x))^3 + (1 - exp(-4*x)*A(x))^4 + (1 - exp(-5*x)*A(x))^5 + ... + (1 - exp(-n*x)*A(x))^n + ...
RELATED SERIES.
log(A(x)) = x + 2*x^2/2! + 30*x^3/3! + 938*x^4/4! + 48030*x^5/5! + 3590522*x^6/6! + 366038190*x^7/7! + 48627434858*x^8/8! + 8147891495550*x^9/9! + 1679856055853402*x^10/10! + ...
Conjecture: after initial term, the coefficients of log(A(x)) modulo 10 has period 4: [2,0,8,0] repeating.
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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-1, (1 - Ser(A)*exp(-m*x +x*O(x^#A)))^m ), #A-1) ); n!*A[n+1]}
for(n=0, 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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