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 = 7; other cases include A380681 (p=2), A380682 (p=3), A380683 (p=5), 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) 7*x = Sum_{n=-oo..+oo} (-1)^n * x^(7*n) * (A(x) + x^n)^(7*n).
(2) 7*x = Sum_{n=-oo..+oo} (-1)^n * x^(7*n*(n-1)) / (1 + x^n*A(x))^(7*n).
EXAMPLE
G.f.: A(x) = 1 + 4*x + 20*x^2 + 110*x^3 + 638*x^4 + 3828*x^5 + 23515*x^6 + 146974*x^7 + 930940*x^8 + 5959555*x^9 + 38485153*x^10 + ...
SPECIFIC VALUES.
A(t) = 4 at t = 0.13350740791591289000322613314614594283415702440618...
A(t) = 15/4 at t = 0.13306070043163671333555422395602596134648327912474...
A(t) = 7/2 at t = 0.13202545577542905621079531887957305202594184840405...
A(t) = 13/4 at t = 0.13034287894006815026999345570114112803185825739759...
A(t) = 3 at t = 0.12791673538949156008130896413813139786694492087583...
A(t) = 11/4 at t = 0.12458285752178691010534021325041522539589331243393...
A(t) = 5/2 at t = 0.12006771822376069162779732794796095008826794384810...
A(t) = 9/4 at t = 0.11392368629179192203472875996890056494098585584228...
A(t) = 2 at t = 0.10541454240800279864304045857992484943305860341173...
A(t) = 7/4 at t = 0.09329415106985017543741414432750112997862686437277...
A(t) = 3/2 at t = 0.07534225027603642407979918399213845227907806585934...
A(t) = 5/4 at t = 0.04729135545134043833540569723928423880850376817601...
A(1/8) = 2.7774160915078697961968184200356959335082725445279...
A(1/9) = 2.1577506229400579652864928021348290523349693083295...
A(1/10) = 1.876798904603102301902233049795776225487607161536...
A(1/11) = 1.710277633713942493378663100908287178830537363898...
A(1/12) = 1.598696964985383318467241544334481791795525744434...
A(1/14) = 1.457254008748038528604246150982429678781363631752...
where 1/2 = Sum_{n=-oo..+oo} (-1)^n * (1/14)^(7*n) * (A(1/14) + (1/14)^n)^(7*n).
PROG
(PARI) {a(n) = my(V=[1]); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef(x - (1/7)*sum(n=-#V, #V, (-1)^n * x^(7*n) * (A + x^n)^(7*n) ), #V) ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 30 2025
STATUS
approved