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Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} x^(n+1) * Product_{k=0..n} (x^k + A(x)).
1

%I #15 Feb 19 2024 08:28:54

%S 1,1,3,6,14,37,95,253,692,1918,5389,15316,43940,127105,370290,1085460,

%T 3199412,9476409,28191145,84195358,252352517,758806718,2288425896,

%U 6920189847,20978805020,63744392583,194101043120,592205499720,1810158252644,5542516035931,16997912424298

%N Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} x^(n+1) * Product_{k=0..n} (x^k + A(x)).

%H Paul D. Hanna, <a href="/A369544/b369544.txt">Table of n, a(n) for n = 1..301</a>

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

%F (1) A(x) = Sum_{n>=0} x^(n+1) * Product_{k=0..n} (x^k + A(x)).

%F (2) A(x) = -1 + Sum_{n>=1} x^(n*(n-1)/2) / Product_{k=1..n} (1 - x^k*A(x)).

%F (3) A(x) = x*(1 + A(x))/(1 + F(1)), where F(n) = -x*(x^n + A(x))/(1 + x*(x^n + A(x)) + F(n+1)), a continued fraction.

%F a(n) ~ c * d^n / n^(3/2), where d = 3.218593300531637233032114... and c = 0.551420536503431682402569... - _Vaclav Kotesovec_, Feb 19 2024

%e G.f.: A(x) = x + x^2 + 3*x^3 + 6*x^4 + 14*x^5 + 37*x^6 + 95*x^7 + 253*x^8 + 692*x^9 + 1918*x^10 + 5389*x^11 + 15316*x^12 + ...

%e where A = A(x) satisfies the following sums

%e (1) A(x) = x*(1 + A) + x^2*(1 + A)*(x + A) + x^3*(1 + A)*(x + A)*(x^2 + A) + x^4*(1 + A)*(x + A)*(x^2 + A)*(x^3 + A) + x^5*(1 + A)*(x + A)*(x^2 + A)*(x^3 + A)*(x^4 + A) + ...

%e and

%e (2) A(x) = -1 + 1/(1 - x*A) + x/((1 - x*A)*(1 - x^2*A)) + x^3/((1 - x*A)*(1 - x^2*A)*(1 - x^3*A)) + x^6/((1 - x*A)*(1 - x^2*A)*(1 - x^3*A)*(1 - x^4*A)) + x^10/((1 - x*A)*(1 - x^2*A)*(1 - x^3*A)*(1 - x^4*A)*(1 - x^5*A)) + ...

%e Further, A = A(x) satisfies the continued fraction

%e (3) A(x) = x*(1 + A)/(1 - x*(x + A)/(1 + x*(x + A) - x*(x^2 + A)/(1 + x*(x^2 + A) - x*(x^3 + A)/(1 + x*(x^3 + A) - x*(x^4 + A)/(1 + x*(x^4 + A) - x*(x^5 + A)/(1 + x*(x^5 + A) - x*(x^6 + A)/(1 + x*(x^6 + A) + ...))))))).

%e SPECIFIC VALUES.

%e A(1/5) = 0.2831219305595313547666016802108274164605432946401360...

%e A(1/4) = 0.4243087446597560528411347345223789877598412481770880...

%e A(2/7) = 0.6005081784072201810613576293832454565671753437307175...

%e A(3/10) = 0.7311141939170660477367006904610649127841162783569496...

%e A(t) = 0.8 at t = 0.30467385134189076808464770503464780231936925568...

%e A(t) = 0.9 at t = 0.30882465867699125967932563981416443155628939160...

%e A(t) = 1 at t = 0.3105475527580721794415140688271783377114855761284799179...

%e A(x) at x = 1/3 diverges.

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = Vec( sum(m=0, #A, x^(m+1) * prod(k=0, m, x^k + x*Ser(A)) ) ) );A[n]}

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

%Y Cf. A229188.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Feb 05 2024