cp's OEIS Frontend

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

%I A379763 #10 Jan 22 2025 03:39:18
%S A379763 1,8,28,80,340,2872,23272,150496,878032,5590352,40944964,308188080,
%T A379763 2214574160,15460447160,109979357264,810265214336,6054587741784,
%U A379763 44971580074120,332187742343988,2466464253968144,18500526368526048,139644462606436800,1055241582609777512,7976465101937086048
%N A379763 G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) + x^n)^(n-1).
%C A379763 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.
%H A379763 Paul D. Hanna, <a href="/A379763/b379763.txt">Table of n, a(n) for n = 0..400</a>
%F A379763 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A379763 (1) 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) + x^n)^(n-1).
%F A379763 (2) 1/2 = Sum_{n=-oo..+oo} x^(n^2) / (1 + x^n*A(x))^(n+1).
%F A379763 a(n) ~ c * d^n / n^(3/2), where d = 8.0740814675... and c = 1.25869706... - _Vaclav Kotesovec_, Jan 22 2025
%e A379763 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 + ...
%e A379763 SPECIFIC VALUES.
%e A379763 A(t) = 4 at t = 0.1235342678268539440746589398189578740264504317462121...
%e A379763 A(t) = 7/2 at t = 0.12337584148360853579899960670632890137087362247055...
%e A379763 A(t) = 3 at t = 0.1189669970336741794074612973623362011930913609542464...
%e A379763 A(t) = 8/3 at t = 0.11236236009985673845496192883838338061075287809042...
%e A379763 A(t) = 5/2 at t = 0.10760338088663649599824099427959331111765863368322...
%e A379763 A(t) = 2 at t = 0.0860421126120690497056915080654929742231128974945892...
%e A379763 A(t) = 5/3 at t = 0.06455762863947182072889129821695321012477178467912...
%e A379763 A(t) = 3/2 at t = 0.05139332682125823774630591999711573636194198482312...
%e A379763 A(t) = 4/3 at t = 0.03643110079983399886726516650416070970893737185267...
%e A379763 A(1/9) = 2.61903290816405002799089092593044410910194535029138...
%e A379763 A(1/10) = 2.2906610607876438864547993548373950931057028357479...
%e A379763 A(1/11) = 2.0918693839543664253067320311652491735792259386896...
%e A379763 A(1/12) = 1.9521586978927587023994157391373410559426298682696...
%e A379763 A(1/16) = 1.6391356345727767379864792642142307766503410761688...
%e A379763 A(1/20) = 1.4835552560753585028949446320205963648290177148078...
%o A379763 (PARI) {a(n) = my(V=[1]); for(i=1,n, V=concat(V,0); A = Ser(V);
%o A379763 V[#V] = polcoef(-2 + 4*sum(n=-#V,#V, x^n * (A + x^n)^(n-1) ),#V-1) );V[n+1]}
%o A379763 for(n=0,30,print1(a(n),", "))
%Y A379763 Cf. A379765.
%K A379763 nonn
%O A379763 0,2
%A A379763 _Paul D. Hanna_, Jan 22 2025