A379763 G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) + x^n)^(n-1).

1, 8, 28, 80, 340, 2872, 23272, 150496, 878032, 5590352, 40944964, 308188080, 2214574160, 15460447160, 109979357264, 810265214336, 6054587741784, 44971580074120, 332187742343988, 2466464253968144, 18500526368526048, 139644462606436800, 1055241582609777512, 7976465101937086048

Offset: 0

Author

Paul D. Hanna, Jan 22 2025

Keywords

• nonn

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds for all y as a formal power series in x.

Examples

			G.f.: A(x) = 1 + 8*x + 28*x^2 + 80*x^3 + 340*x^4 + 2872*x^5 + 23272*x^6 + 150496*x^7 + 878032*x^8 + 5590352*x^9 + 40944964*x^10 + ...
SPECIFIC VALUES.
A(t) = 4 at t = 0.1235342678268539440746589398189578740264504317462121...
A(t) = 7/2 at t = 0.12337584148360853579899960670632890137087362247055...
A(t) = 3 at t = 0.1189669970336741794074612973623362011930913609542464...
A(t) = 8/3 at t = 0.11236236009985673845496192883838338061075287809042...
A(t) = 5/2 at t = 0.10760338088663649599824099427959331111765863368322...
A(t) = 2 at t = 0.0860421126120690497056915080654929742231128974945892...
A(t) = 5/3 at t = 0.06455762863947182072889129821695321012477178467912...
A(t) = 3/2 at t = 0.05139332682125823774630591999711573636194198482312...
A(t) = 4/3 at t = 0.03643110079983399886726516650416070970893737185267...
A(1/9) = 2.61903290816405002799089092593044410910194535029138...
A(1/10) = 2.2906610607876438864547993548373950931057028357479...
A(1/11) = 2.0918693839543664253067320311652491735792259386896...
A(1/12) = 1.9521586978927587023994157391373410559426298682696...
A(1/16) = 1.6391356345727767379864792642142307766503410761688...
A(1/20) = 1.4835552560753585028949446320205963648290177148078...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A379765.

Programs

PARI

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

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

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

a(n) ~ c * d^n / n^(3/2), where d = 8.0740814675... and c = 1.25869706... - Vaclav Kotesovec, Jan 22 2025