A378264 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (1 + A(x)^n)^(n+1).

1, 3, 10, 38, 164, 783, 4005, 21400, 117602, 659019, 3748736, 21588796, 125646501, 737977155, 4369147468, 26048215099, 156249597852, 942344615209, 5710710976884, 34756875588376, 212361179832431, 1302068876523950, 8009024360554817, 49407447276951470, 305609996146288873, 1895015255546957578

Offset: 1

Author

Paul D. Hanna, Dec 08 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 + 3*x^2 + 10*x^3 + 38*x^4 + 164*x^5 + 783*x^6 + 4005*x^7 + 21400*x^8 + 117602*x^9 + 659019*x^10 + 3748736*x^11 + 21588796*x^12 + ...
SPECIFIC VALUES.
A(t) = 1/3 at t = 0.14832728317680424382350400745104642263167027946862...
A(t) = 1/4 at t = 0.13433913917600443178696714330960568436967435856815...
A(t) = 1/5 at t = 0.12029812285398972879219940261295281978412524937754...
A(3/20) = 0.3521325903099608361455770617898033111722103407971...
A(1/7) = 0.29252723487814042698570516039406838227427731852655...
A(1/8) = 0.21500724214149512130643660913381998900575603076452...
A(1/9) = 0.17407688053908806913569913139334508111874650183559...
A(1/10) = 0.14711097488062849474543678333471254427936118296317...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..366

Crossrefs

Cf. A379200, A379199, A166952, 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*(1 + A^m)^(m+1) ), #V-3); ); polcoef(A,n)}
for(n=1,40,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 * (1 + A(x)^n)^(n+1).

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

From Paul D. Hanna, Dec 20 2024: (Start)

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

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

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

(End)