A379199 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n - 1)^(n+1).

1, 1, 2, 2, 4, 9, 45, 164, 546, 1493, 3944, 10588, 32997, 112945, 396404, 1330461, 4265180, 13292275, 41778612, 135378928, 452828655, 1534394542, 5175561385, 17246318586, 56998526633, 188492707958, 628391304843, 2115131897264, 7162685531894, 24280930956521, 82152859633099

Offset: 1

Author

Paul D. Hanna, Dec 20 2024

Keywords

• nonn

Comments

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

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 45*x^7 + 164*x^8 + 546*x^9 + 1493*x^10 + 3944*x^11 + 10588*x^12 + ...
SPECIFIC VALUES.
A(t) = 1/2 at t = 0.28045847462385815185359630099816126187110099265378...
  where t = 1/Sum_{n=-oo..+oo} (-1)^n * (2^(n-1) - 1)^n / 2^(n^2-1),
  also, t = 1/Sum_{n=-oo..+oo} (2^(n-1) + 1)^(n-1) / 2^(n^2-1).
A(t) = 1/3 at t = 0.23482705460970305955617199360925350115096428519729...
  where t = 1/Sum_{n=-oo..+oo} (-1)^n * (3^(n-1) - 1)^n / 3^(n^2-1),
  also, t = 1/Sum_{n=-oo..+oo} (3^(n-1) + 1)^(n-1) / 3^(n^2-1).
A(t) = 1/4 at t = 0.19291797602834900465339136778069433360676297133766...
  where t = 1/Sum_{n=-oo..+oo} (-1)^n * (4^(n-1) - 1)^n / 4^(n^2-1),
  also, t = 1/Sum_{n=-oo..+oo} (4^(n-1) + 1)^(n-1) / 4^(n^2-1).
A(1/4) = 0.37094847513809700088242935848658292140487254454012...
  where 4 = Sum_{n=-oo..+oo} A(1/4)^n * (A(1/4)^n - 1)^(n+1),
  also, 4 = Sum_{n=-oo..+oo} A(1/4)^(2*n) * (A(1/4)^n + 1)^n.
A(1/5) = 0.26269124124750053890427847522296583687631694884657...
A(1/6) = 0.20631303406093749454201994379654348907240460444958...
A(1/7) = 0.17034902087146833005156413354158308643804109633470...
A(1/8) = 0.14521334319041207588863463072178319621820854479438...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..301

Crossrefs

Cf. A379200, A166952, A378264, A379202, A379203, A379204, A379205.

Programs

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, A^m*(A^m - 1)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))

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)^n * (A(x)^n - 1)^(n+1).

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

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

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

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