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%I #6 Mar 02 2025 22:51:01
%S 1,1,4,14,44,130,496,1586,5128,17764,59492,196368,659330,2226166,
%T 7396070,24876724,83420692,279644938,935867180,3146178556,10534161782,
%U 35369902036,118498115768,398015733448,1333108657368,4477017033638,15004173961698,50369493608278,168842274387828,566766393991544
%N G.f. satisfies x = Sum_{n>=1} -(-1)^(n mod 3) * x^n * abs(1/A(x)^n), where abs(F(x)) equals the series expansion formed by the unsigned coefficients in F(x).
%C a(n) ~ c*d^n, where d = 3.354756161514554082... and c = 0.322808122437862772...
%H Paul D. Hanna, <a href="/A381354/b381354.txt">Table of n, a(n) for n = 0..400</a>
%e G.f.: A(x) = 1 + x + 4*x^2 + 14*x^3 + 44*x^4 + 130*x^5 + 496*x^6 + 1586*x^7 + 5128*x^8 + 17764*x^9 + 59492*x^10 + 196368*x^11 + 659330*x^12 + ...
%e The coefficients in 1/A(x)^n begin
%e n = 1: [1, -1, -3, -7, -11, -5, -87, -131, -111, ...];
%e n = 2: [1, -2, -5, -8, 1, 54, -49, 96, 753, ...];
%e n = 3: [1, -3, -6, -4, 27, 129, -72, 132, 1041, ...];
%e n = 4: [1, -4, -6, 4, 59, 184, -260, -168, 749, ...];
%e n = 5: [1, -5, -5, 15, 90, 194, -650, -670, 675, ...];
%e n = 6: [1, -6, -3, 28, 114, 144, -1226, -1068, 1851, ...];
%e n = 7: [1, -7, 0, 42, 126, 28, -1932, -974, 5131, ...];
%e n = 8: [1, -8, 4, 56, 122, -152, -2684, 8, 10915, ...];
%e n = 9: [1, -9, 9, 69, 99, -387, -3381, 2223, 18990, ...];
%e ...
%e The unsigned coefficients form the series abs(1/A(x)^n) and begin
%e n = 1: [1, 1, 3, 7, 11, 5, 87, 131, 111, ...];
%e n = 2: [1, 2, 5, 8, 1, 54, 49, 96, 753, ...];
%e n = 3: [1, 3, 6, 4, 27, 129, 72, 132, 1041, ...];
%e n = 4: [1, 4, 6, 4, 59, 184, 260, 168, 749, ...];
%e n = 5: [1, 5, 5, 15, 90, 194, 650, 670, 675, ...];
%e n = 6: [1, 6, 3, 28, 114, 144, 1226, 1068, 1851, ...];
%e n = 7: [1, 7, 0, 42, 126, 28, 1932, 974, 5131, ...];
%e n = 8: [1, 8, 4, 56, 122, 152, 2684, 8, 10915, ...];
%e n = 9: [1, 9, 9, 69, 99, 387, 3381, 2223, 18990, ...];
%e ...
%e The g.f. A(x) causes the following sum to equal x:
%e x = Sum_{n>=1} -(-1)^(n mod 3) * x^n * abs(1/A(x)^n), i.e.,
%e x = x *(1 + 1*x + 3*x^2 + 7*x^3 + 11*x^4 + 5*x^5 + ...)
%e - x^2*(1 + 2*x + 5*x^2 + 8*x^3 + 1*x^4 + 54*x^5 + ...)
%e - x^3*(1 + 3*x + 6*x^2 + 4*x^3 + 27*x^4 + 129*x^5 + ...)
%e + x^4*(1 + 4*x + 6*x^2 + 4*x^3 + 59*x^4 + 184*x^5 + ...)
%e - x^5*(1 + 5*x + 5*x^2 + 15*x^3 + 90*x^4 + 194*x^5 + ...)
%e - x^6*(1 + 6*x + 3*x^2 + 28*x^3 + 114*x^4 + 144*x^5 + ...)
%e + x^7*(1 + 7*x + 0*x^2 + 42*x^3 + 126*x^4 + 28*x^5 + ...)
%e - x^8*(1 + 8*x + 4*x^2 + 56*x^3 + 122*x^4 + 152*x^5 + ...)
%e - x^9*(1 + 9*x + 9*x^2 + 69*x^3 + 99*x^4 + 387*x^5 + ...)
%e + ...
%e SPECIFIC VALUES.
%e A(t) = 5/2 at t = 0.2439659541767366435324668966...
%e A(t) = 9/4 at t = 0.2353840551855704278278595775153205...
%e A(t) = 2 at t = 0.223632028538443643064216621211748830622323...
%e A(t) = 7/4 at t = 0.20657376416781114780239812337080772813821570967375...
%e A(t) = 5/3 at t = 0.19903893798837337323032062934473873273603265809256...
%e A(t) = 3/2 at t = 0.17957316117018144667020305457516566265734157029382...
%e A(t) = 4/3 at t = 0.15062899618726958291222230561587359930435166593485...
%e A(t) = 5/4 at t = 0.13003050002674430753834725398705188137096165805154...
%e A(1/4) = 2.7283787868137581335275349...
%e A(1/5) = 1.6765917121780458577176419255019373572188147733104...
%e A(1/6) = 1.4165864667091498858294378983473697621659718923072...
%e A(1/7) = 1.2992396855197346868111927874895540857809847001779...
%e A(1/8) = 1.2327343901263073476621403289881692603672322742418...
%o (PARI) {a(n) = my(V=[1]); for(i=0,n, V=concat(V,'t); A=Ser(V);
%o V[#V] = 't - polcoef(-x + sum(m=1,#A+2, -(-1)^(m%3) * x^m * Ser(abs(Vec(1/A^m))) ), #V)); polcoef(H=A,n)}
%o for(n=0,30,print1(a(n),", "))
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 02 2025