A367390 - OEIS

%I #20 Jan 29 2024 11:58:13

%S 1,2,9,52,545,6366,98707,1700840,35405505,817958170,21500633891,

%T 618661892652,19636408658737,675144805723766,25147073628948195,

%U 1004734122294047056,42965745214637476097,1955039747566085781426,94404335950307686644163,4818562790963397438214100

%N Expansion of e.g.f. A(x) satisfying A(x)^2 = exp(x) * A(x*A(x)) with A(0) = 0.

%C Note that if F(x)^2 = exp(x) * F(x*F(x)) with F(0) = 1, then F(x) is the e.g.f. of A367391.

%H Paul D. Hanna, <a href="/A367390/b367390.txt">Table of n, a(n) for n = 1..310</a>

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

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

%F Let B^n(x) denote the n-th iteration of B(x) = x*A(x), where B^(n+1)(x) = B( B^n(x) ) with B^0(x) = x, then

%F (2) log( A(x)/x ) = Sum_{n>=0} B^n(x).

%F (3) B^n(x) = x*A(x)^(2^n - 1) / exp( Sum_{k=0..n-2} (2^(n-k-1) - 1) * B^k(x) ) for n > 1.

%F (3.a) B^2(x) = x*A(x)^3 / exp(x).

%F (3.b) B^3(x) = x*A(x)^7 / exp(3*x + B(x)).

%F (3.c) B^4(x) = x*A(x)^15 / exp(7*x + 3*B(x) + B^2(x)).

%F (3.d) B^5(x) = x*A(x)^31 / exp(15*x + 7*B(x) + 3*B^2(x) + B^3(x)).

%F (4) A( B^n(x) ) = A(x)^(2^n) / exp( Sum_{k=0..n-1} 2^(n-k-1) * B^k(x) ) for n > 0.

%F (4.a) A(B(x)) = A(x)^2 / exp(x).

%F (4.b) A(B^2(x)) = A(x)^4 / exp(2*x + B(x)).

%F (4.c) A(B^3(x)) = A(x)^8 / exp(4*x + 2*B(x) + B^2(x)).

%F (4.d) A(B^4(x)) = A(x)^16 / exp(8*x + 4*B(x) + 2*B^2(x) + B^3(x)).

%e E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 545*x^5/5! + 6366*x^6/6! + 98707*x^7/7! + 1700840*x^8/8! + 35405505*x^9/9! + 817958170*x^10/10! + ...

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

%e A(x)^2 = 2*x^2/2! + 12*x^3/3! + 96*x^4/4! + 880*x^5/5! + 11280*x^6/6! + 167664*x^7/7! + 3030944*x^8/8! + ...

%e A(x*A(x)) = 2*x^2/2! + 6*x^3/3! + 60*x^4/4! + 500*x^5/5! + 7230*x^6/6! + 104202*x^7/7! + 1962296*x^8/8! + ...

%e Let B(x) = x*A(x), then log( A(x)/x ) equals the sum of all iterations of B(x)

%e log( A(x)/x ) = x + B(x) + B(B(x)) + B(B(B(x))) + B(B(B(B(x)))) + ...

%e which is equivalent to

%e log( A(x)/x ) = x + x*A(x) + x*A(x)*A(x*A(x)) + x*A(x)*A(x*A(x)) * A( x*A(x)*A(x*A(x)) ) + ...

%e RELATED SERIES.

%e A(x)/x = 1 + x + 3*x^2/2! + 13*x^3/3! + 109*x^4/4! + 1061*x^5/5! + 14101*x^6/6! + 212605*x^7/7! + 3933945*x^8/8! + 81795817*x^9/9! + ...

%e log( A(x)/x ) = x + 2*x^2/2! + 6*x^3/3! + 60*x^4/4! + 500*x^5/5! + 6870*x^6/6! + 96642*x^7/7! + 1824536*x^8/8! + 36995688*x^9/9! + ...

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

%e B(x) = 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 3270*x^6/6! + 44562*x^7/7! + 789656*x^8/8! + ...

%e B(B(x)) = 24*x^4/4! + 240*x^5/5! + 3600*x^6/6! + 52080*x^7/7! + 994560*x^8/8! + ...

%e B(B(B(x))) = 40320*x^8/8! + 1451520*x^9/9! + 50803200*x^10/10! + ...

%e B(B(B(B(x)))) = 20922789888000*x^16/16! + 2845499424768000*x^17/17! + ...

%e etc.

%e where A(x) = x * exp(x + B(x) + B(B(x)) + B(B(B(x))) + B(B(B(B(x)))) + ...).

%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) - exp(-x +x*O(x^(#V)))*A^2, #V) ); n!*V[n+1]}

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

%Y Cf. A367391, A144681, A367386, A367387.

%Y Cf. A369090.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 08 2024