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A213219
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Largest k such that F = 1 + x*(x*F^n)'/F^k is solved by a power series F with positive coefficients (where ' is the derivative w.r.t. x).
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1
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1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107, 108, 110, 112, 113, 115, 116, 118, 119, 121, 122, 124, 125, 127, 129
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OFFSET
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1,2
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COMMENTS
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It can be shown that it does not make a difference if "positive" is replaced by "nonnegative": The coefficients of the power series are either increasing (possibly constant) from the initial 1 on (so they are strictly positive), or they start being strictly decreasing from a given rank on (so one coefficient may equal zero and all following ones will be negative).
Can someone explain and/or find an analytic expression for the limit a(n)/n -> 1.5419... (cf. formula)?
Related functions: F(x) = 1 + x*(x*F(x)^n)'/F(x)^(n-1) is satisfied by F(x) = Sum_{m>=0} x^m*Product_{k=0..m-1} (n*k+1), the g.f. of n-tuple factorials. - Paul D. Hanna, Mar 03 2013
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LINKS
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FORMULA
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lim_{n->oo} a(n)/n = 1.5419041168209403448882049905180407467654... [calculated by Paul D. Hanna, Mar 03 2013]
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EXAMPLE
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For n=5, k=7, the equation is solved by F = 1 + x + 3*x^2 + 12*x^3 + 65*x^4 + ... (only positive coefficients), for k=8 the solution is F = 1 + x + 2*x^2 - 56*x^4 - 833*x^5 - ...(negative coefficients), therefore a(5)=7.
For powers of 10, we have:
n a(n)
10^0 1
10^1 14
10^2 153
10^3 1541
10^4 15418
10^5 154189
10^6 1541903
10^7 15419040
10^8 154190411
10^9 1541904116
10^10 15419041167
10^11 154190411681
10^12 1541904116820
10^13 15419041168208
10^14 154190411682093
10^15 1541904116820939
10^16 15419041168209402
10^17 154190411682094033
10^18 1541904116820940344
10^19 15419041168209403448
10^20 154190411682094034488
10^21 1541904116820940344887
10^22 15419041168209403448881
10^23 154190411682094034488819
10^24 1541904116820940344888204
10^25 15419041168209403448882049
10^26 154190411682094034488820498
10^27 1541904116820940344888204990
10^28 15419041168209403448882049904
10^29 154190411682094034488820499051
10^30 1541904116820940344888204990517
10^31 15419041168209403448882049905179
10^32 154190411682094034488820499051803
...
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PROG
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(PARI) A213219(n)={my(F(n, k, m=15)=my(o=O(x^m), G=1+x); until(G==G=1+x*deriv(x*G^n, x)/G^k+o, ); G); for(k=n, 2*n, vecmin(Vec(F(n, k)))>0 || return(k-1))}
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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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