OFFSET
0,2
COMMENTS
Conjecture: if F(x) satisfies p*x = Sum_{n=-oo..+oo} (-1)^n * x^(p*n) * (F(x) + x^n)^(p*n) with F(0) = 1 for some fixed integer p, then F(x) is an integer series in x iff p is prime. The g.f. A(x) of this sequence is the case when p = 5; other cases include A380681 (p=2), A380682 (p=3), A380684 (p=7), A380685 (p=11), and A380686 (p=13).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(5*n) * (A(x) + x^n)^(5*n).
(2) 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(5*n*(n-1)) / (1 + x^n*A(x))^(5*n).
EXAMPLE
G.f.: A(x) = 1 + 3*x + 11*x^2 + 44*x^3 + 185*x^4 + 806*x^5 + 3620*x^6 + 16732*x^7 + 79540*x^8 + 388643*x^9 + 1949282*x^10 + ...
SPECIFIC VALUES.
A(t) = 3 at t = 0.1656585561542188229179039339279667279544762240304050...
A(t) = 14/5 at t = 0.1644890266657217057337777551090303278709118268002...
A(t) = 11/4 at t = 0.1639940862845506446584856440134731375010152643849...
A(t) = 8/3 at t = 0.16296555715018582608065911469825172623459454694320...
A(t) = 5/2 at t = 0.16005707095639639434280252651911632827450237245277...
A(t) = 7/3 at t = 0.15583134086682481297371758918910187148728620506691...
A(t) = 9/4 at t = 0.15314543706542281624067397763769279965083824277452...
A(t) = 2 at t = 0.1423022206763674557195760443937635836494945667701269...
A(t) = 7/4 at t = 0.12600854660327543342909005578232040563632497923962...
A(t) = 5/3 at t = 0.11894865283164989456553421133445906732777245081931...
A(t) = 3/2 at t = 0.10152500411277069852130955132141456974567990788237...
A(t) = 4/3 at t = 0.07817396758638139366536098908292028967434502027552...
PROG
(PARI) {a(n) = my(V=[1]); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef(x - (1/5)*sum(n=-#V, #V, (-1)^n * x^(5*n) * (A + x^n)^(5*n) ), #V) ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Paul D. Hanna, Jan 30 2025
STATUS
approved