A377103 G.f. A(x) satisfies Sum_{n>=1} A(x^n - 3*x^(n+1)) = x.

1, 2, 14, 108, 932, 8494, 80816, 792834, 7964766, 81525244, 847234316, 8915840154, 94819534880, 1017494297362, 11003391627478, 119797392972660, 1312015757405828, 14444656015113630, 159773654937327260, 1774697222194801818, 19787293416104986024, 221379967241156347006, 2484547366889151931430

Offset

1,2

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 - 3*x^(n+1)).

(2) x = Sum_{n>=1} a(n) * x^n * (1-3*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 - 12*x))/6 = C(3*x)/3 and C(x) is the g.f. of the Catalan numbers (A000108).

a(n) ~ c * 12^n / n^(3/2), where c = 0.041106239119057419763719574770353578634816297455230079187498... - Vaclav Kotesovec, Oct 26 2024

Example

G.f.: A(x) = x + 2*x^2 + 14*x^3 + 108*x^4 + 932*x^5 + 8494*x^6 + 80816*x^7 + 792834*x^8 + 7964766*x^9 + 81525244*x^10 + 847234316*x^11 + 8915840154*x^12 + ...
where
x = A(x - 3*x^2) + A(x^2 - 3*x^3) + A(x^3 - 3*x^4) + A(x^4 - 3*x^5) + A(x^5 - 3*x^6) + A(x^6 - 3*x^7) + ...
Also,
x = a(1)*x*(1-3*x)/(1-x) + a(2)*x^2*(1-3*x)^2/(1-x^2) + a(3)*x^3*(1-3*x)^3/(1-x^3) + a(4)*x^4*(1-3*x)^4/(1-x^4) + ...
SPECIFIC VALUES.
Note that x = Sum_{n>=1} A(x^n - 2*x^(n+1)) holds for -1/15 <= x <= 1/6.
1/6 = A(3/6^2) + A(3/6^3) + A(3/6^4) + A(3/6^5) + A(3/6^6) + ...
1/7 = A(4/7^2) + A(4/7^3) + A(4/7^4) + A(4/7^5) + A(4/7^6) + ...
1/8 = A(5/8^2) + A(5/8^3) + A(5/8^4) + A(5/8^5) + A(5/8^6) + ...
-1/15 = A(-2/25) + A(18/15^3) + A(-18/15^4) + A(18/15^5) + ...
A(1/12) = 0.14956091765123027740377899281470645932919267413067...
A(4/49) = 0.12894954654418769297020015649231204633105229586338...
A(5/64) = 0.11363138588219220039269772014857100093492862921410...
A(2/27) = 0.10170632771991407843228415767466801054005630940554...
A(-2/25) = -0.071726024357260297922935430546055655984452853890...

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 - 3*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*(1-3*x)^m/(1-x^m +O(x^#V))), #V-1) ); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A377101, A377102, A000108.

Sequence in context: A103945 A378460 A111713 * A144278 A359108 A214766

Adjacent sequences: A377100 A377101 A377102 * A377104 A377105 A377106

Keyword

nonn

Author

Paul D. Hanna, Oct 18 2024

Status

approved