This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378573 #10 Dec 11 2024 09:27:38
%S A378573 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,
%T A378573 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,
%U A378573 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
%N A378573 G.f. A(x) = Sum_{n=-oo..+oo} x^n * (1 + x^(3*n+1))^(2*n).
%C A378573 Related identities:
%C A378573 (C.1) Sum_{n=-oo..+oo} x^n * (1 - x^(3*n+2))^n = 0.
%C A378573 (C.2) Sum_{n=-oo..+oo} (-1)^n * x^(3*n*(n+1)) / (1 + x^(3*n+1))^(n+1) = 0.
%H A378573 Paul D. Hanna, <a href="/A378573/b378573.txt">Table of n, a(n) for n = 0..8200</a>
%F A378573 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A378573 (1.a) A(x) = Sum_{n=-oo..+oo} x^n * (1 + x^(3*n+1))^(2*n).
%F A378573 (1.b) A(x) = Sum_{n=-oo..+oo} x^n * (1 - x^(3*n+1))^(2*n).
%F A378573 (2.a) A(x) = Sum_{n=-oo..+oo} x^(3*n*(2*n-1)) / (1 + x^(3*n-1))^(2*n).
%F A378573 (2.b) A(x) = Sum_{n=-oo..+oo} x^(3*n*(2*n-1)) / (1 - x^(3*n-1))^(2*n).
%F A378573 (3.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (1 + x^(3*n+2))^n.
%F A378573 (3.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} (-1)^n * x^n * (1 - x^(3*n+2))^n.
%F A378573 (4.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(3*n*(n+1)) / (1 + x^(3*n+1))^(n+1).
%F A378573 (4.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(3*n*(n+1)) / (1 - x^(3*n+1))^(n+1).
%F A378573 (5.a) A(x^2) = Sum_{n=-oo..+oo} x^(2*n-1) * (1 + x^(6*n-1))^(2*n-1).
%F A378573 (5.b) A(x^2) = -Sum_{n=-oo..+oo} x^(2*n-1) * (1 - x^(6*n-1))^(2*n-1).
%F A378573 (6.a) A(x^2) = Sum_{n=-oo..+oo} x^(6*n*(2*n+1)) / (1 + x^(6*n+1))^(2*n+1).
%F A378573 (6.b) A(x^2) = Sum_{n=-oo..+oo} x^(6*n*(2*n+1)) / (1 - x^(6*n+1))^(2*n+1).
%e A378573 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 + ...
%e A378573 SPECIFIC VALUES.
%e A378573 A(z) = 0 at z = -0.6726473467784327964946394402158022892169850805633511277...
%e A378573 where 0 = Sum_{n=-oo..+oo} z^n * (1 + z^(3*n+1))^(2*n).
%e A378573 A(t) = 8 at t = 0.80674137409155594738508715662274076269252097031895...
%e A378573 A(t) = 7 at t = 0.79012273526862596166723863415319642411267133718829...
%e A378573 A(t) = 6 at t = 0.76819406763538484112048712466638978472377443909212...
%e A378573 A(t) = 5 at t = 0.73777899222025289918616588954081072720456797874020...
%e A378573 A(t) = 4 at t = 0.69251246918071024586564098631094327512630629569865...
%e A378573 A(t) = 3 at t = 0.61751935356221793541340938213050415525442896744761...
%e A378573 A(t) = 2 at t = 0.46815043155347172312205584241722605840913217439574...
%e A378573 where 2 = Sum_{n=-oo..+oo} t^n * (1 + t^(3*n+1))^(2*n).
%e A378573 A(t) = -1 at t = -0.77517567890012104592411512614387150563591857093990...
%e A378573 A(t) = -2 at t = -0.81854774961928757410155043510790044331007733543405...
%e A378573 a(t) = -3 at t = -0.84450708995424907597930320956281576983716334202613...
%e A378573 A(4/5) = 7.5631342681464228254307790972990507013398446615499...
%e A378573 A(3/4) = 5.3598446737980629233504982857608095266005392614233...
%e A378573 A(2/3) = 3.5884575039557777965471951540301884270744196834272...
%e A378573 A(3/5) = 2.8340662386949213795469239985484973660637713412934...
%e A378573 A(1/2) = 2.1531614564039262021396751059639076614616014159933...
%e A378573 where A(1/2) = Sum_{n=-oo..+oo} (2^(3*n+1) + 1)^(2*n) / 2^(3*n*(2*n+1)).
%e A378573 A(2/5) = 1.7360641537921941524470509621075633132346504795101...
%e A378573 A(1/3) = 1.5384884473879487136671091866260679901472537410267...
%e A378573 where A(1/3) = Sum_{n=-oo..+oo} (3^(3*n+1) + 1)^(2*n) / 3^(3*n*(2*n+1)).
%e A378573 A(1/4) = 1.3491464535561504459384378489663997806149645910211...
%e A378573 where A(1/4) = Sum_{n=-oo..+oo} (4^(3*n+1) + 1)^(2*n) / 4^(3*n*(2*n+1)).
%e A378573 A(1/5) = 1.2580390146694337862857122246948093151986010024710...
%e A378573 A(-1/3) = 0.71151183654243437744829702888914217294561469541322...
%e A378573 A(-1/2) = 0.51369607391129963764388587069816692146820022971981...
%e A378573 A(-2/3) = 0.03171944560249247956083537535107529561830519903038...
%o A378573 (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)}
%o A378573 for(n=0,100, print1(a(n),", "))
%Y A378573 Cf. A260147.
%K A378573 nonn
%O A378573 0,4
%A A378573 _Paul D. Hanna_, Dec 10 2024