OFFSET
1,2
COMMENTS
Suggested by Paul D. Hanna, cf. link.
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
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..1000
Paul D. Hanna, New? Constant for a Functional Equation, SeqFan list, Mar 2 2013.
FORMULA
lim_{n->oo} a(n)/n = 1.5419041168209403448882049905180407467654... [calculated by Paul D. Hanna, Mar 03 2013]
EXAMPLE
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
...
PROG
(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))}
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Mar 02 2013
STATUS
approved