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A367386 Expansion of g.f. A(x) satisfying A(x)^2 = (1+x) * A(x*A(x)) with A(0) = 0.

%I A367386 #20 Jan 08 2024 15:30:28
%S A367386 1,1,1,2,3,6,12,26,56,124,278,632,1454,3378,7918,18694,44427,106175,
%T A367386 255031,615320,1490588,3624013,8840006,21628173,53061676,130508716,
%U A367386 321743567,794907220,1967848545,4880622339,12125865713,30175562392,75207082211,187707922818,469126856364
%N A367386 Expansion of g.f. A(x) satisfying A(x)^2 = (1+x) * A(x*A(x)) with A(0) = 0.
%C A367386 Note that if F(x)^2 = (1+x) * F(x*F(x)) with F(0) = 1, then F(x) is the g.f. of A120056.
%H A367386 Paul D. Hanna, <a href="/A367386/b367386.txt">Table of n, a(n) for n = 1..972</a>
%F A367386 G.f. A(x) = Sum_{n>=1} a(n)*x^n and B(x) = x*A(x) satisfy the following formulas.
%F A367386 (1) A(x)^2 = (1+x) * A(x*A(x)).
%F A367386 (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 A367386 The iterations of B(x) = x*A(x) begin
%F A367386 (3.a) B(B(x)) = x*A(x)^3 / (1+x).
%F A367386 (3.b) B(B(B(x))) = x*A(x)^7 / ((1+x)^3 * (1 + x*A(x))).
%F A367386 (3.c) B(B(B(B(x)))) = x*A(x)^15 / ((1+x)^7 * (1 + x*A(x))^3 * (1 + x*A(x)^3/(1+x))).
%F A367386 (3.d) B(B(B(B(B(x))))) = x*A(x)^31 / ((1+x)^15 * (1+x*A(x))^7 * (1 + x*A(x)^3/(1+x))^3 * (1 + x*A(x)^7/((1+x)^3*(1+x*A(x))))).
%F A367386 The compositions of g.f. A(x) with the iterations of B(x) = x*A(x) begin
%F A367386 (4.a) A(B(x)) = A(x)^2 / (1+x).
%F A367386 (4.b) A(B(B(x))) = A(x)^4 / ((1+x)^2 * (1 + x*A(x))).
%F A367386 (4.c) A(B(B(B(x)))) = A(x)^8 / ((1+x)^4 * (1 + x*A(x))^2 * (1 + x*A(x)^3/(1+x))).
%F A367386 (4.d) A(B(B(B(B(x))))) = A(x)^16 / ((1+x)^8 * (1+x*A(x))^4 * (1 + x*A(x)^3/(1+x))^2 * (1 + x*A(x)^7/((1+x)^3*(1+x*A(x))))).
%e A367386 G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 12*x^7 + 26*x^8 + 56*x^9 + 124*x^10 + 278*x^11 + 632*x^12 + 1454*x^13 + 3378*x^14 + 7918*x^15 + ...
%e A367386 where A(x)^2 = (1+x) * A(x*A(x)) as can be seen from the following expansions
%e A367386 A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 11*x^6 + 22*x^7 + 46*x^8 + 100*x^9 + 221*x^10 + 496*x^11 + 1128*x^12 + 2592*x^13 + 6016*x^14 + 14080*x^15 + ...
%e A367386 A(x*A(x)) = x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 15*x^7 + 31*x^8 + 69*x^9 + 152*x^10 + 344*x^11 + 784*x^12 + 1808*x^13 + 4208*x^14 + 9872*x^15 + ...
%e A367386 Let B(x) = x*A(x), then A(x) equals the infinite product involving successive iterations of B(x) starting with
%e A367386 A(x) = x*(1+x) * (1 + B(x)) * (1 + B(B(x))) * (1 + B(B(B(x)))) * (1 + B(B(B(B(x))))) * ...
%e A367386 which is equivalent to
%e A367386 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 A367386 RELATED SERIES.
%e A367386 Successive iterations of B(x) = x*A(x) begin
%e A367386 B(x) = x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 6*x^7 + 12*x^8 + 26*x^9 + 56*x^10 + ...
%e A367386 B(B(x)) = x^4 + 2*x^5 + 4*x^6 + 9*x^7 + 18*x^8 + 39*x^9 + 85*x^10 + 191*x^11 + ...
%e A367386 B(B(B(x))) = x^8 + 4*x^9 + 12*x^10 + 34*x^11 + 89*x^12 + 228*x^13 + 575*x^14 + ...
%e A367386 B(B(B(B(x)))) = x^16 + 8*x^17 + 40*x^18 + 164*x^19 + 594*x^20 + 1984*x^21 + ...
%e A367386 B(B(B(B(B(x))))) = x^32 + 16*x^33 + 144*x^34 + 968*x^35 + 5412*x^36 + 26592*x^37 + ...
%e A367386 etc.
%e A367386 The coefficients in the iterations of x*A(x) form a table that begins
%e A367386 n=1: [1, 1, 1, 2, 3, 6, 12, 26, 56, 124, 278, 632, 1454, ...];
%e A367386 n=2: [1, 2, 4, 9, 18, 39, 85, 191, 433, 994, 2303, 5377, ...];
%e A367386 n=3: [1, 4, 12, 34, 89, 228, 575, 1441, 3595, 8943, 22215, ...];
%e A367386 n=4: [1, 8, 40, 164, 594, 1984, 6266, 19006, 55944, 160926, ...];
%e A367386 n=5: [1, 16, 144, 968, 5412, 26592, 118692, 491820, 1920852, ...];
%e A367386 n=6: [1, 32, 544, 6544, 62536, 505152, 3584008, 22917912, ...];
%e A367386 n=7: [1, 64, 2112, 47904, 839824, 12132480, 150360848, ...];
%e A367386 n=8: [1, 128, 8320, 366144, 12271904, 334108928, 7695888928, ...];
%e A367386 n=9: [1, 256, 33024, 2862208, 187499072, 9902461440, ...];
%e A367386 n=10: [1, 512, 131584, 22632704, 2931033216, 304847561728, ...];
%e A367386 etc.
%o A367386 (PARI) {a(n) = my(A=x, V=[0,1]); for(i=1,n, V = concat(V,0); A = Ser(V);
%o A367386 V[#V] = polcoeff( subst(A,x,x*A) - A^2/(1+x), #V) ); V[n+1]}
%o A367386 for(n=1,40, print1(a(n),", "))
%Y A367386 Cf. A120056, A367387, A367390.
%K A367386 nonn
%O A367386 1,4
%A A367386 _Paul D. Hanna_, Jan 08 2024