A377102 G.f. A(x) satisfies Sum_{n>=1} A(x^n - 2*x^(n+1)) = x.

1, 1, 5, 26, 153, 937, 5991, 39388, 264892, 1813345, 12595527, 88550874, 628917019, 4505821155, 32525365501, 236331968980, 1727163960917, 12687375261788, 93626506061591, 693765269582186, 5159886375484379, 38506369969370543, 288244195882614691, 2163770393221878704, 16284902775963901362

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..600

Formula

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

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

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

(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).

a(n) ~ c * 8^n / n^(3/2), where c = 0.054055052368862534473343088562219044348670873006004998292... - Vaclav Kotesovec, Oct 26 2024

Example

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 + ...
where
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) + ...
Also,
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) + ...
SPECIFIC VALUES.
Note that x = Sum_{n>=1} A(x^n - 2*x^(n+1)) holds for -1/10 <= x <= 1/4.
1/4 = A(2/4^2) + A(2/4^3) + A(2/4^4) + A(2/4^5) + A(2/4^6) + ...
1/5 = A(3/5^2) + A(3/5^3) + A(3/5^4) + A(3/5^5) + A(3/5^6) + ...
1/6 = A(4/6^2) + A(4/6^3) + A(4/6^4) + A(4/6^5) + A(4/6^6) + ...
-1/10 = A(-3/25) + A(12/10^3) + A(-12/10^4) + A(12/10^5) + ...
A(1/8) = 0.20710610711842392242527905050300334682088922121239...
A(3/25) = 0.1693202521828569395690775728560326265563621878016...
A(1/9) = 0.14405637354944947221857025390412263100357065268950...
A(5/49) = 0.1256165981161689692118876811902510058791832035549...
A(1/10) = 0.1219518894988437087641183777164139114069464382555...
A(3/32) = 0.1114590446546772577032226835138763611866212819438...
A(-3/25) = -0.0877809989044440033318394249428874864709938244825...

Prog

(PARI) \\ using formula (1)
{a(n) = my(V=[0, 1]); for(i=1, n, V=concat(V, 0); A=Ser(V);
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)}
for(n=1, 30, print1(a(n), ", "))
(PARI) \\ using formula (2)
{a(n) = my(V=[0, 1]); for(i=1, n, V=concat(V, 0); A=Ser(V);
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)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A377101, A377103, A000108.

Sequence in context: A263134 A082029 A365150 * A081047 A001705 A185108

Adjacent sequences: A377099 A377100 A377101 * A377103 A377104 A377105

Keyword

nonn

Author

Paul D. Hanna, Oct 18 2024

Status

approved