A378573 G.f. A(x) = Sum_{n=-oo..+oo} x^n * (1 + x^(3*n+1))^(2*n).

1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 6, 1, 1, 1, 8, 7, 1, 2, 10, 1, 1, 1, 27, 1, 1, 1, 14, 11, 1, 30, 16, 1, 1, 1, 18, 1, 46, 36, 20, 1, 1, 1, 37, 67, 2, 1, 24, 85, 1, 1, 117, 1, 1, 1, 28, 71, 1, 286, 30, 1, 22, 1, 33, 1, 154, 1, 34, 287, 211, 1, 36, 191, 1, 1, 38, 1, 127, 456, 271, 1, 1, 524, 42, 2, 1, 277, 44, 681, 1, 1, 46, 1, 788, 1, 1049

Offset: 0

Author

Paul D. Hanna, Dec 10 2024

Keywords

• nonn

Comments

Related identities:

(C.1) Sum_{n=-oo..+oo} x^n * (1 - x^(3*n+2))^n = 0.

(C.2) Sum_{n=-oo..+oo} (-1)^n * x^(3*n*(n+1)) / (1 + x^(3*n+1))^(n+1) = 0.

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + x^4 + x^5 + x^6 + 4*x^7 + x^8 + 2*x^9 + x^10 + 6*x^11 + x^12 + x^13 + x^14 + 8*x^15 + 7*x^16 + x^17 + 2*x^18 + 10*x^19 + x^20 + ...
SPECIFIC VALUES.
A(z) = 0 at z = -0.6726473467784327964946394402158022892169850805633511277...
  where 0 = Sum_{n=-oo..+oo} z^n * (1 + z^(3*n+1))^(2*n).
A(t) = 8 at t = 0.80674137409155594738508715662274076269252097031895...
A(t) = 7 at t = 0.79012273526862596166723863415319642411267133718829...
A(t) = 6 at t = 0.76819406763538484112048712466638978472377443909212...
A(t) = 5 at t = 0.73777899222025289918616588954081072720456797874020...
A(t) = 4 at t = 0.69251246918071024586564098631094327512630629569865...
A(t) = 3 at t = 0.61751935356221793541340938213050415525442896744761...
A(t) = 2 at t = 0.46815043155347172312205584241722605840913217439574...
  where 2 = Sum_{n=-oo..+oo} t^n * (1 + t^(3*n+1))^(2*n).
A(t) = -1 at t = -0.77517567890012104592411512614387150563591857093990...
A(t) = -2 at t = -0.81854774961928757410155043510790044331007733543405...
a(t) = -3 at t = -0.84450708995424907597930320956281576983716334202613...
A(4/5) = 7.5631342681464228254307790972990507013398446615499...
A(3/4) = 5.3598446737980629233504982857608095266005392614233...
A(2/3) = 3.5884575039557777965471951540301884270744196834272...
A(3/5) = 2.8340662386949213795469239985484973660637713412934...
A(1/2) = 2.1531614564039262021396751059639076614616014159933...
  where A(1/2) = Sum_{n=-oo..+oo} (2^(3*n+1) + 1)^(2*n) / 2^(3*n*(2*n+1)).
A(2/5) = 1.7360641537921941524470509621075633132346504795101...
A(1/3) = 1.5384884473879487136671091866260679901472537410267...
  where A(1/3) = Sum_{n=-oo..+oo} (3^(3*n+1) + 1)^(2*n) / 3^(3*n*(2*n+1)).
A(1/4) = 1.3491464535561504459384378489663997806149645910211...
  where A(1/4) = Sum_{n=-oo..+oo} (4^(3*n+1) + 1)^(2*n) / 4^(3*n*(2*n+1)).
A(1/5) = 1.2580390146694337862857122246948093151986010024710...
A(-1/3) = 0.71151183654243437744829702888914217294561469541322...
A(-1/2) = 0.51369607391129963764388587069816692146820022971981...
A(-2/3) = 0.03171944560249247956083537535107529561830519903038...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..8200

Crossrefs

Cf. A260147.

Programs

PARI

{a(n) = my(A = sum(m=-n,n, x^m * (1 + x^(3*m+1) +x*O(x^n))^(2*m) )); polcoef(A,n)}
for(n=0,100, print1(a(n),", "))

Formula

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

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

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

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

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

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

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

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

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

(5.a) A(x^2) = Sum_{n=-oo..+oo} x^(2*n-1) * (1 + x^(6*n-1))^(2*n-1).

(5.b) A(x^2) = -Sum_{n=-oo..+oo} x^(2*n-1) * (1 - x^(6*n-1))^(2*n-1).

(6.a) A(x^2) = Sum_{n=-oo..+oo} x^(6*n*(2*n+1)) / (1 + x^(6*n+1))^(2*n+1).

(6.b) A(x^2) = Sum_{n=-oo..+oo} x^(6*n*(2*n+1)) / (1 - x^(6*n+1))^(2*n+1).