A380682 G.f. A(x) satisfies 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (A(x) + x^n)^(3*n) with A(0) = 1.

1, 2, 5, 16, 61, 249, 1052, 4563, 20235, 91420, 419423, 1948712, 9149749, 43345282, 206922865, 994442783, 4807332168, 23360888569, 114048745449, 559114028719, 2751327072141, 13585135755721, 67287432632212, 334223723285970, 1664459140656762, 8309074228762074, 41571618824043394, 208418278965591903

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 = 3; other cases include A380681 (p=2), A380683 (p=5), A380684 (p=7), A380685 (p=11), and A380686 (p=13).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (A(x) + x^n)^(3*n).

(2) 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(3*n*(n-1)) / (1 + x^n*A(x))^(3*n).

Example

G.f.: A(x) = 1 + 2*x + 5*x^2 + 16*x^3 + 61*x^4 + 249*x^5 + 1052*x^6 + 4563*x^7 + 20235*x^8 + 91420*x^9 + 419423*x^10 + ...
SPECIFIC VALUES.
A(t) = 5/2 at t = 0.188871427312089002362445972530648971415711424302944...
  where 3*t = Sum_{n=-oo..+oo} (-1)^n * t^(3*n) * (5/2 + t^n)^(3*n).
A(t) = 12/5 at t = 0.18868910633609162854751746221130590766164638340904...
A(t) = 7/3 at t = 0.188310811613049359264872709286394200394007556585091...
A(t) = 9/4 at t = 0.187491961572287107936822145509769093278573633191499...
A(t) = 2 at t = 0.18191184646831190821977457143548097497140466434055881...
  where 3*t = Sum_{n=-oo..+oo} (-1)^n * t^(3*n) * (2 + t^n)^(3*n).
A(t) = 7/4 at t = 0.168981723006713284104403940380385366604996062747880...
A(t) = 5/3 at t = 0.162153325115107190099315913212808038512641198435824...
A(t) = 3/2 at t = 0.142934334890975216226788769727414735079216818663752...
A(t) = 4/3 at t = 0.113325137338106856353715539630517619876361458642495...
A(t) = 5/4 at t = 0.093117742165138705834667869876172949210337355112453...
A(t) = 6/5 at t = 0.078809295361620649265962846705221894786693261346932...
A(1/6) = 1.7195575688646614941709649951426444498574451702352...
  where 1/2 = Sum_{n=-oo..+oo} (-1)^n * (1/6)^(3*n) * (A(1/6) + (1/6)^n)^(3*n).
A(1/7) = 1.4994581495647669286957761240856093881831880400378...
A(1/8) = 1.3908912823194582799072892122892423088223072225884...
A(1/9) = 1.3233036505403952069617510047004525961407492658012...
A(1/10) = 1.276470986170846283229728384247703521445790888233...
A(1/12) = 1.215146100179149322027534682517117886109324924788...
  where 1/4 = Sum_{n=-oo..+oo} (-1)^n * (1/12)^(3*n) * (A(1/12) + (1/12)^n)^(3*n).

Prog

(PARI) {a(n) = my(V=[1]); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef(x - (1/3)*sum(n=-#V, #V, (-1)^n * x^(3*n) * (A + x^n)^(3*n) ), #V) ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A380681, A380683, A380684, A380685, A380686.

Sequence in context: A012051 A012159 A009736 * A349458 A370797 A307228

Adjacent sequences: A380679 A380680 A380681 * A380683 A380684 A380685

Keyword

nonn

Author

Paul D. Hanna, Jan 30 2025

Status

approved