A378585 - OEIS

%I #7 Dec 14 2024 07:08:51

%S 1,-1,2,-7,24,-90,345,-1373,5610,-23418,99373,-427370,1858665,

%T -8160629,36123158,-161033302,722322008,-3257737848,14764170412,

%U -67202964003,307090370840,-1408254347355,6478788726514,-29893940649690,138306656545279,-641475676394960,2982029118960410

%N G.f. A(x) satisfies Sum_{n=-oo..+oo} x^n * (1 - A(x)^n)^(n+2) = 0.

%H Paul D. Hanna, <a href="/A378585/b378585.txt">Table of n, a(n) for n = 1..400</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) 0 = Sum_{n=-oo..+oo} x^n * (1 - A(x)^n)^(n+2).

%F (2) 0 = Sum_{n=-oo..+oo, n<>0} x^n * A(x)^(n*(n-2)) / (1 - A(x)^n)^(n-2).

%e G.f.: A(x) = x - x^2 + 2*x^3 - 7*x^4 + 24*x^5 - 90*x^6 + 345*x^7 - 1373*x^8 + 5610*x^9 - 23418*x^10 + 99373*x^11 - 427370*x^12 + ...

%e where 0 = Sum_{n=-oo..+oo} x^n * (1 - A(x)^n)^(n+2).

%e SPECIFIC VALUES.

%e A(t) = -1/3 at t = -0.202048598781426028974428756010099171051365847414101...

%e   where 0 = Sum_{n=-oo..+oo} t^n * (1 - (-1/3)^n)^(n+2).

%e A(t) = 1/6 at t = 0.196484473181370501094128491316425714189080003441911...

%e   where 0 = Sum_{n=-oo..+oo} t^n * (1 - 1/6^n)^(n+2).

%e A(t) = 1/7 at t = 0.164302639314666689556062074167390219500364417716145...

%e A(t) = 1/8 at t = 0.141209865117619967564257913696969927533119492860642...

%e A(t) = 1/9 at t = 0.123812084870245998422013539668902476299512367653013...

%e A(1/5) = 0.169192343778061922209842951936810267398473367614055...

%e   where 0 = Sum_{n=-oo..+oo} (1/5)^n * (1 - A(1/5)^n)^(n+2).

%e A(1/6) = 0.144648560513959037971202801780478849604291701771996...

%e A(1/7) = 0.126295474717177983310858620199338125046642276703407...

%e A(1/8) = 0.112071559142787952670745084130795493860180444964021...

%e A(-1/5) = -0.317384435332550315318178922121258411565438673373317...

%e   where 0 = Sum_{n=-oo..+oo} (-1/5)^n * (1 - A(-1/5)^n)^(n+2).

%e A(-1/6) = -0.218143303309150197595640133851028719055081320361627...

%e A(-1/7) = -0.175217297452511685103517475866596161617743122001224...

%e A(-1/8) = -0.147651296038267940756784606949037889432459534245007...

%o (PARI) {a(n) = my(V=[0,1],A); for(i=1,n, V=concat(V,0); A=Ser(V);

%o V[#V] = polcoef( -sum(m=-#A,#A, x^m*(1 - A^m)^(m+2) ), #V-4) ); polcoef(A,n)}

%o for(n=1,30, print1(a(n),", "))

%K sign

%O 1,3

%A _Paul D. Hanna_, Dec 13 2024