A378572 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n + 2)^(2*n+1).

1, 1, 5, 14, 89, 371, 2161, 10696, 61969, 335434, 1952772, 11086400, 65242174, 381110396, 2268552339, 13498467880, 81190422545, 489403577479, 2970230729380, 18078456241060, 110557602461243, 678051379830676, 4173297748385770, 25753520575460880, 159374541532819470, 988579243982587598

Offset

1,3

Comments

Related identity: Sum_{n=-oo..+oo} x^(2*n) * (y - x^n)^(n+1) = 0 which holds formally for all y.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..520

Formula

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

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

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

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

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

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

Example

G.f.: A(x) = x + x^2 + 5*x^3 + 14*x^4 + 89*x^5 + 371*x^6 + 2161*x^7 + 10696*x^8 + 61969*x^9 + 335434*x^10 + 1952772*x^11 + 11086400*x^12 + ...
If B(A(x)) = x, then B(x) = 1/Sum_{n=-oo..+oo} x^(2*n) * (x^n + 2)^(2*n+1), where
1/B(x) = 1/x + 1 + 4*x + x^2 + 28*x^3 + 65*x^5 + 104*x^6 + 256*x^7 + 1473*x^9 + 280*x^10 + 4096*x^11 + 2304*x^12 + 16384*x^13 + 1793*x^14 + 77420*x^15 + ...
so that 1/x = 1/A(x) + 1 + 4*A(x) + A(x)^2 + 28*A(x)^3 + 65*A(x)^5 + 104*A(x)^6 + 256*A(x)^7 + 1473*A(x)^9 + 280*A(x)^10 + 4096*A(x)^11 + 2304*A(x)^12 + ...
SPECIFIC VALUES.
A(t) = 1/4 at t = 0.15124391075754253639558588965972746377899271778087...
  where t = 1/Sum_{n=-oo..+oo} (2*4^n + 1)^(2*n+1) / 4^(n*(2*n+3)),
  also t = -1/Sum_{n=-oo..+oo} (2*4^n - 1)^(2*n+1) / 4^(n*(2*n+3)).
A(t) = 1/5 at t = 0.14093189935284192112853624800184111153721092989398...
  where t = 1/Sum_{n=-oo..+oo} (2*5^n + 1)^(2*n+1) / 5^(n*(2*n+3)),
  also t = -1/Sum_{n=-oo..+oo} (2*5^n - 1)^(2*n+1) / 5^(n*(2*n+3)).
A(t) = 1/6 at t = 0.12762034386259473890208447519583555495225930857270...
  where t = 1/Sum_{n=-oo..+oo} (2*6^n + 1)^(2*n+1) / 6^(n*(2*n+3)),
  also t = -1/Sum_{n=-oo..+oo} (2*6^n - 1)^(2*n+1) / 6^(n*(2*n+3)).
A(t) = 1/7 at t = 0.11522633540905433103213355006598004706615043040042...
A(1/7) = 0.20619537625303988294154964608060409441061086391029...
  where 7 = Sum_{n=-oo..+oo} A(1/7)^(2*n) * (A(1/7)^n + 2)^(2*n+1).
A(1/8) = 0.16124084635791323271390159552368535440362428871416...
A(1/9) = 0.13578822593330510423411549603521083003433094538543...
A(1/10) = 0.1181262074118177682352533737455449237598984970039...
A(-1/7) = -0.135011909725363875083802383229181183889341951058...

Prog

(PARI) N=30 \\ number of terms
{a(n) = my(R = sum(m=-N-1, N+1, x^(2*m) * (x^m + 2 +x^2*O(x^N))^(2*m+1) ), A);
A = serreverse(1/R); polcoef(A, n)}
for(n=1, N, print1(a(n), ", "))

Crossrefs

Cf. A378571.

Sequence in context: A198039 A198091 A197797 * A224245 A374510 A238380

Adjacent sequences: A378569 A378570 A378571 * A378573 A378574 A378575

Keyword

nonn

Author

Paul D. Hanna, Dec 30 2024

Status

approved