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 = 11; other cases include A380681 (p=2), A380682 (p=3), A380683 (p=5), A380684 (p=7), 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) 11*x = Sum_{n=-oo..+oo} (-1)^n * x^(11*n) * (A(x) + x^n)^(11*n).
(2) 11*x = Sum_{n=-oo..+oo} (-1)^n * x^(11*n*(n-1)) / (1 + x^n*A(x))^(11*n).
EXAMPLE
G.f.: A(x) = 1 + 6*x + 46*x^2 + 391*x^3 + 3519*x^4 + 32844*x^5 + 314364*x^6 + 3065049*x^7 + 30309929*x^8 + 303099290*x^9 + 3058547381*x^10 + ...
SPECIFIC VALUES.
A(t) = 6 at t = 0.090000476553610098333936561993113431230179656035951...
A(t) = 23/4 at t = 0.089904362552717018496281212379042887341660822182720...
A(t) = 11/2 at t = 0.089762108143955999804906680510431963722502315631017...
A(t) = 21/4 at t = 0.089571784785331406017933955922771923839577204671422...
A(t) = 5 at t = 0.089328048916016148567266915549556681404921507326294...
A(t) = 19/4 at t = 0.089021828930896167930729300018052241545262626534003...
A(t) = 9/2 at t = 0.088639727012093783278631270443300123555082726212816...
A(t) = 17/4 at t = 0.088163008492574729328946144487211928464698491854612...
A(t) = 4 at t = 0.087566004568351523753674557036812324986540207491711...
A(t) = 15/4 at t = 0.086813659723044456115942277751839295606362337512786...
A(t) = 7/2 at t = 0.085857786661170026686815721066549191640656363954370...
A(t) = 13/4 at t = 0.084631294361884811609106602966070064213388886060655...
A(t) = 3 at t = 0.083039118725675563873572063063947631966896957743615...
A(t) = 11/4 at t = 0.080943581668291716513955743656344894368200272615840...
A(t) = 5/2 at t = 0.078139939225219767004436168156945342731806997142928...
A(t) = 9/4 at t = 0.074313822111290058468037790952705269913914160734189...
A(t) = 2 at t = 0.068963358485741297538930072936707964684506643542405...
A(t) = 7/4 at t = 0.061247664018491269786368757076755916988911184384664...
A(t) = 3/2 at t = 0.049668605948759164521143823094160539257693482555411...
A(t) = 4/3 at t = 0.038474881471047206740938456038121405771468073277438...
A(t) = 5/4 at t = 0.031333035603003913799823189445750544049757434319706...
PROG
(PARI) {a(n) = my(V=[1]); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef(x - (1/11)*sum(n=-#V, #V, (-1)^n * x^(11*n) * (A + x^n)^(11*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