A378584 - OEIS

%I #10 Jan 22 2025 04:13:35

%S 1,2,5,13,46,170,666,2648,10944,46296,199828,873616,3863994,17255100,

%T 77710773,352486026,1608824138,7383210030,34048297506,157700809314,

%U 733283992543,3421740870626,16018406425836,75208586777464,354067037861768,1671014124932980,7904396487947504,37469496390403704

%N G.f. A(x) satisfies 1 + 4*A(x) = Sum_{n=-oo..+oo} (2*x + A(x)^n)^n.

%H Paul D. Hanna, <a href="/A378584/b378584.txt">Table of n, a(n) for n = 1..380</a>

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

%F (1) 1 + 4*A(x) = Sum_{n=-oo..+oo} (2*x + A(x)^n)^n.

%F (2) 1 + 4*A(x) = Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 2*x*A(x)^n)^n.

%F a(n) ~ c * d^n / n^(3/2), where d = 5.00995688191010718208... and c = 0.138126946089315438... - _Vaclav Kotesovec_, Jan 22 2025

%e G.f: A(x) = x + 2*x^2 + 5*x^3 + 13*x^4 + 46*x^5 + 170*x^6 + 666*x^7 + 2648*x^8 + 10944*x^9 + 46296*x^10 + 199828*x^11 + 873616*x^12 + ...

%e where 1 + 4*A(x) = Sum_{n=-oo..+oo} (2*x + A(x)^n)^n.

%e SPECIFIC VALUES.

%e A(t) = 1/3 at t = 0.1831955345652769491608413185059304103972018821511469491...

%e   where 7/3 = Sum_{n=-oo..+oo} (2*t + 1/3^n)^n,

%e   also, 7/3 = Sum_{n=-oo..+oo} (1/3)^(n^2) / (1 + 2*t/3^n)^n.

%e A(t) = 1/4 at t = 0.1588173428481951756613569181105876136100886668873381763...

%e   where 2 = Sum_{n=-oo..+oo} (2*t + 1/4^n)^n,

%e   also, 2 = Sum_{n=-oo..+oo} (1/4)^(n^2) / (1 + 2*t/4^n)^n.

%e A(1/6) = 0.272948495724025612756651865489...

%e A(1/7) = 0.20982707763947888749753415117948193411561266617062...

%e A(1/8) = 0.17189390051289055911618566206048257447724135382709...

%e A(1/10) = 0.1270422272370836396738203332383911172335941544364...

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

%o V[#V] = (1/2)*polcoef( sum(n=-#V,#V, (2*x + A^n)^n ) - 4*A, #V-1) ); V[n+1]}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A378583.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 10 2025