A377102 - OEIS

%I #19 Oct 26 2024 06:51:22

%S 1,1,5,26,153,937,5991,39388,264892,1813345,12595527,88550874,

%T 628917019,4505821155,32525365501,236331968980,1727163960917,

%U 12687375261788,93626506061591,693765269582186,5159886375484379,38506369969370543,288244195882614691,2163770393221878704,16284902775963901362

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

%H Paul D. Hanna, <a href="/A377102/b377102.txt">Table of n, a(n) for n = 1..600</a>

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

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

%F (2) x = Sum_{n>=1} a(n) * x^n * (1-2*x)^n/(1-x^n).

%F (3) F(x) = Sum_{n>=1} a(n) * x^n / (1 - F(x)^n), where F(x) = (1 - sqrt(1 - 8*x))/4 = C(2*x)/2 and C(x) is the g.f. of the Catalan numbers (A000108).

%F a(n) ~ c * 8^n / n^(3/2), where c = 0.054055052368862534473343088562219044348670873006004998292... - _Vaclav Kotesovec_, Oct 26 2024

%e G.f.: A(x) = x + x^2 + 5*x^3 + 26*x^4 + 153*x^5 + 937*x^6 + 5991*x^7 + 39388*x^8 + 264892*x^9 + 1813345*x^10 + 12595527*x^11 + 88550874*x^12 + ...

%e where

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

%e Also,

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

%e SPECIFIC VALUES.

%e Note that x = Sum_{n>=1} A(x^n - 2*x^(n+1)) holds for -1/10 <= x <= 1/4.

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

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

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

%e -1/10 = A(-3/25) + A(12/10^3) + A(-12/10^4) + A(12/10^5) + ...

%e A(1/8) = 0.20710610711842392242527905050300334682088922121239...

%e A(3/25) = 0.1693202521828569395690775728560326265563621878016...

%e A(1/9) = 0.14405637354944947221857025390412263100357065268950...

%e A(5/49) = 0.1256165981161689692118876811902510058791832035549...

%e A(1/10) = 0.1219518894988437087641183777164139114069464382555...

%e A(3/32) = 0.1114590446546772577032226835138763611866212819438...

%e A(-3/25) = -0.0877809989044440033318394249428874864709938244825...

%o (PARI) \\ using formula (1)

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

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

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

%o (PARI) \\ using formula (2)

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

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

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

%Y Cf. A377101, A377103, A000108.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Oct 18 2024