A370539 Expansion of g.f. (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x), where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

1, 2, 8, 30, 118, 462, 1824, 7208, 28558, 113274, 449848, 1787968, 7111716, 28303548, 112700032, 448939744, 1788990454, 7131191202, 28433681832, 113398298336, 452345641820, 1804739556100, 7201621713568, 28741559322464, 114722405784428, 457971605148996, 1828422022584176

Offset

0,2

Links

Table of n, a(n) for n=0..26.

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) A(x) = (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x), where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

(2) A(x) = (1 + sqrt(1 - 4*x)) * (2-3*x + x*sqrt(1 - 4*x^2)) / (4*(1-4*x)).

a(n) ~ (10 + sqrt(3)) * 2^(2*n - 5). - Vaclav Kotesovec, Mar 14 2024

Example

G.f.: A(x) = 1 + 2*x + 8*x^2 + 30*x^3 + 118*x^4 + 462*x^5 + 1824*x^6 + 7208*x^7 + 28558*x^8 + 113274*x^9 + 449848*x^10 + ...
RELATED SERIES.
The Catalan function C(x) = (1 - sqrt(1-4*x))/(2*x) begins
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + ... + A000108(n)*x^n + ...

Prog

(PARI) {a(n) = my(x = 'x + O('x^(n+3)), C(x) = (1 - sqrt(1 - 4*x))/(2*x), A = (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) );
polcoeff(A, n); }
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(x = 'x + O('x^(n+3)), A = (1 + sqrt(1 - 4*x)) * sqrt( (1 - 2*x)*(1 - sqrt(1 - 4*x^2))/2 ) / (2*x*(1-4*x)) );
polcoeff(A, n); }
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A000108, A370540.

Sequence in context: A266319 A155116 A133915 * A150759 A150760 A230588

Adjacent sequences: A370536 A370537 A370538 * A370540 A370541 A370542

Keyword

nonn

Author

Paul D. Hanna, Mar 12 2024

Status

approved