A367387 Expansion of g.f. A(x) satisfying A(x)^2 = A(x*A(x)) / (1-x) with A(0) = 0.

1, 1, 2, 3, 7, 14, 34, 77, 193, 472, 1214, 3099, 8122, 21293, 56666, 151261, 407519, 1102006, 2998716, 8189515, 22467935, 61841586, 170818016, 473173219, 1314463002, 3660532769, 10218207713, 28584456783, 80124502593, 225011930357, 633003693094, 1783658958681, 5033641233827

Offset

1,3

Comments

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.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..938

Formula

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

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

(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).

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

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

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

(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.

(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.

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

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

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

(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.

(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.

Example

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 + ...
where A(x)^2 = A(x*A(x)) / (1-x) as can be seen from the following expansions
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 + ...
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 + ...
Let B(x) = x*A(x), then A(x) equals the infinite product involving successive iterations of B(x) starting with
A(x) = x/(1-x) / ( (1 - B(x)) * (1 - B(B(x))) * (1 - B(B(B(x)))) * (1 - B(B(B(B(x))))) * ...)
which is equivalent to
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)))) * ...).
RELATED SERIES.
Successive iterations of B(x) = x*A(x) begin
B(x) = x^2 + x^3 + 2*x^4 + 3*x^5 + 7*x^6 + 14*x^7 + 34*x^8 + 77*x^9 + ...
B(B(x)) = x^4 + 2*x^5 + 6*x^6 + 13*x^7 + 35*x^8 + 84*x^9 + 221*x^10 + ...
B(B(B(x))) = x^8 + 4*x^9 + 16*x^10 + 50*x^11 + 159*x^12 + 470*x^13 + ...
B(B(B(B(x)))) = x^16 + 8*x^17 + 48*x^18 + 228*x^19 + 974*x^20 + 3812*x^21 + ...
B(B(B(B(B(x))))) = x^32 + 16*x^33 + 160*x^34 + 1224*x^35 + 7900*x^36 + ...
etc.
The coefficients in the iterations of x*A(x) form a table that begins
n=1: [1, 1, 2, 3, 7, 14, 34, 77, 193, 472, 1214, 3099, ...];
n=2: [1, 2, 6, 13, 35, 84, 221, 556, 1464, 3801, 10107, ...];
n=3: [1, 4, 16, 50, 159, 470, 1397, 4033, 11656, 33284, ...];
n=4: [1, 8, 48, 228, 974, 3812, 14142, 50182, 172562, ...];
n=5: [1, 16, 160, 1224, 7900, 45096, 234764, 1136732, ...];
n=6: [1, 32, 576, 7568, 80568, 734672, 5938776, ...];
n=7: [1, 64, 2176, 52000, 977264, 15344032, 208985520, ...];
n=8: [1, 128, 8448, 382528, 13345504, 382081856, ...];
n=9: [1, 256, 33280, 2927744, 195986880, 10643805824, ...];
n=10: [1, 512, 132096, 22894848, 2998537088, 316503534848, ...];
etc.

Prog

(PARI) {a(n) = my(A=x, V=[0, 1]); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoeff( subst(A, x, x*A) - (1-x)*A^2, #V) ); V[n+1]}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Cf. A088792, A367386, A367390.

Sequence in context: A035083 A328057 A305785 * A185089 A180564 A113822

Adjacent sequences: A367384 A367385 A367386 * A367388 A367389 A367390

Keyword

nonn

Author

Paul D. Hanna, Jan 08 2024

Status

approved