%I #14 Jan 08 2024 15:35:27

%S 1,1,2,3,7,14,34,77,193,472,1214,3099,8122,21293,56666,151261,407519,

%T 1102006,2998716,8189515,22467935,61841586,170818016,473173219,

%U 1314463002,3660532769,10218207713,28584456783,80124502593,225011930357,633003693094,1783658958681,5033641233827

%N Expansion of g.f. A(x) satisfying A(x)^2 = A(x*A(x)) / (1-x) with A(0) = 0.

%C Note that if F(x)^2 = (1+x) * F(x*F(x)) with F(0) = 1, then F(x) is the g.f. of A088792.

%H Paul D. Hanna, <a href="/A367387/b367387.txt">Table of n, a(n) for n = 1..938</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n and B(x) = x*A(x) satisfies the following formulas.

%F (1) A(x)^2 = A(x*A(x)) / (1-x).

%F (2) A(x) = x/(1-x) / ( (1 - B(x)) * (1 - B(B(x))) * (1 - B(B(B(x)))) * (1 - B(B(B(B(x))))) * ...), an infinite product involving iterations of B(x) = x*A(x).

%F The iterations of B(x) = x*A(x) begin

%F (3.a) B(B(x)) = x*(1-x) * A(x)^3.

%F (3.b) B(B(B(x))) = x*(1-x)^3 * (1 - x*A(x)) * A(x)^7.

%F (3.c) B(B(B(B(x)))) = x*(1-x)^7 * (1 - x*A(x))^3 * (1 - x*(1-x)*A(x)^3) * A(x)^15.

%F (3.d) B(B(B(B(B(x))))) = x*(1-x)^15 * (1 - x*A(x))^7 * (1 - x*(1-x)*A(x)^3)^3 * (1 - x*(1-x)^3*(1-x*A(x))*A(x)^7) * A(x)^31.

%F The compositions of g.f. A(x) with the iterations of B(x) = x*A(x) begin

%F (4.a) A(B(x)) = (1-x) * A(x)^2.

%F (4.b) A(B(B(x))) = (1-x)^2 * (1 - x*A(x)) * A(x)^4.

%F (4.c) A(B(B(B(x)))) = (1-x)^4 * (1 - x*A(x))^2 * (1 - x*(1-x)*A(x)^3) * A(x)^8.

%F (4.d) A(B(B(B(B(x))))) = (1-x)^8 * (1 - x*A(x))^4 * (1 - x*(1-x)*A(x)^3)^2 * (1 - x*(1-x)^3*(1-x*A(x))*A(x)^7) * A(x)^16.

%e G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 7*x^5 + 14*x^6 + 34*x^7 + 77*x^8 + 193*x^9 + 472*x^10 + 1214*x^11 + 3099*x^12 + 8122*x^13 + 21293*x^14 + 56666*x^15 + ...

%e where A(x)^2 = A(x*A(x)) / (1-x) as can be seen from the following expansions

%e A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 10*x^5 + 24*x^6 + 54*x^7 + 133*x^8 + 320*x^9 + 809*x^10 + 2038*x^11 + 5278*x^12 + 13702*x^13 + 36144*x^14 + 95758*x^15 + ...

%e A(x*A(x)) = x^2 + x^3 + 3*x^4 + 5*x^5 + 14*x^6 + 30*x^7 + 79*x^8 + 187*x^9 + 489*x^10 + 1229*x^11 + 3240*x^12 + 8424*x^13 + 22442*x^14 + 59614*x^15 + ...

%e Let B(x) = x*A(x), then A(x) equals the infinite product involving successive iterations of B(x) starting with

%e A(x) = x/(1-x) / ( (1 - B(x)) * (1 - B(B(x))) * (1 - B(B(B(x)))) * (1 - B(B(B(B(x))))) * ...)

%e which is equivalent to

%e A(x) = x*(1-x) / ( (1 - x*A(x)) * (1 - x*A(x) * A(x*A(x))) * (1 - x*A(x) * A(x*A(x)) * A(x*A(x) * A(x*A(x)))) * ...).

%e RELATED SERIES.

%e Successive iterations of B(x) = x*A(x) begin

%e B(x) = x^2 + x^3 + 2*x^4 + 3*x^5 + 7*x^6 + 14*x^7 + 34*x^8 + 77*x^9 + ...

%e B(B(x)) = x^4 + 2*x^5 + 6*x^6 + 13*x^7 + 35*x^8 + 84*x^9 + 221*x^10 + ...

%e B(B(B(x))) = x^8 + 4*x^9 + 16*x^10 + 50*x^11 + 159*x^12 + 470*x^13 + ...

%e B(B(B(B(x)))) = x^16 + 8*x^17 + 48*x^18 + 228*x^19 + 974*x^20 + 3812*x^21 + ...

%e B(B(B(B(B(x))))) = x^32 + 16*x^33 + 160*x^34 + 1224*x^35 + 7900*x^36 + ...

%e etc.

%e The coefficients in the iterations of x*A(x) form a table that begins

%e n=1: [1, 1, 2, 3, 7, 14, 34, 77, 193, 472, 1214, 3099, ...];

%e n=2: [1, 2, 6, 13, 35, 84, 221, 556, 1464, 3801, 10107, ...];

%e n=3: [1, 4, 16, 50, 159, 470, 1397, 4033, 11656, 33284, ...];

%e n=4: [1, 8, 48, 228, 974, 3812, 14142, 50182, 172562, ...];

%e n=5: [1, 16, 160, 1224, 7900, 45096, 234764, 1136732, ...];

%e n=6: [1, 32, 576, 7568, 80568, 734672, 5938776, ...];

%e n=7: [1, 64, 2176, 52000, 977264, 15344032, 208985520, ...];

%e n=8: [1, 128, 8448, 382528, 13345504, 382081856, ...];

%e n=9: [1, 256, 33280, 2927744, 195986880, 10643805824, ...];

%e n=10: [1, 512, 132096, 22894848, 2998537088, 316503534848, ...];

%e etc.

%o (PARI) {a(n) = my(A=x, V=[0,1]); for(i=1,n, V = concat(V,0); A = Ser(V);

%o V[#V] = polcoeff( subst(A,x,x*A) - (1-x)*A^2, #V) ); V[n+1]}

%o for(n=1,40, print1(a(n),", "))

%Y Cf. A088792, A367386, A367390.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jan 08 2024