A378585 G.f. A(x) satisfies Sum_{n=-oo..+oo} x^n * (1 - A(x)^n)^(n+2) = 0.

1, -1, 2, -7, 24, -90, 345, -1373, 5610, -23418, 99373, -427370, 1858665, -8160629, 36123158, -161033302, 722322008, -3257737848, 14764170412, -67202964003, 307090370840, -1408254347355, 6478788726514, -29893940649690, 138306656545279, -641475676394960, 2982029118960410

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..400

Formula

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

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

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

Example

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 + ...
where 0 = Sum_{n=-oo..+oo} x^n * (1 - A(x)^n)^(n+2).
SPECIFIC VALUES.
A(t) = -1/3 at t = -0.202048598781426028974428756010099171051365847414101...
  where 0 = Sum_{n=-oo..+oo} t^n * (1 - (-1/3)^n)^(n+2).
A(t) = 1/6 at t = 0.196484473181370501094128491316425714189080003441911...
  where 0 = Sum_{n=-oo..+oo} t^n * (1 - 1/6^n)^(n+2).
A(t) = 1/7 at t = 0.164302639314666689556062074167390219500364417716145...
A(t) = 1/8 at t = 0.141209865117619967564257913696969927533119492860642...
A(t) = 1/9 at t = 0.123812084870245998422013539668902476299512367653013...
A(1/5) = 0.169192343778061922209842951936810267398473367614055...
  where 0 = Sum_{n=-oo..+oo} (1/5)^n * (1 - A(1/5)^n)^(n+2).
A(1/6) = 0.144648560513959037971202801780478849604291701771996...
A(1/7) = 0.126295474717177983310858620199338125046642276703407...
A(1/8) = 0.112071559142787952670745084130795493860180444964021...
A(-1/5) = -0.317384435332550315318178922121258411565438673373317...
  where 0 = Sum_{n=-oo..+oo} (-1/5)^n * (1 - A(-1/5)^n)^(n+2).
A(-1/6) = -0.218143303309150197595640133851028719055081320361627...
A(-1/7) = -0.175217297452511685103517475866596161617743122001224...
A(-1/8) = -0.147651296038267940756784606949037889432459534245007...

Prog

(PARI) {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A=Ser(V);
V[#V] = polcoef( -sum(m=-#A, #A, x^m*(1 - A^m)^(m+2) ), #V-4) ); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A383573 A052705 A150391 * A150392 A150393 A150394

Adjacent sequences: A378582 A378583 A378584 * A378586 A378587 A378588

Keyword

sign

Author

Paul D. Hanna, Dec 13 2024

Status

approved