A055395 Number of bracketings of 0#0#0#...#0 giving result 0, where 0#0 = 0#1 = 1#0 = 1, 1#1 = 0.

1, 0, 0, 1, 4, 12, 36, 116, 392, 1350, 4696, 16500, 58572, 209824, 757440, 2752185, 10057636, 36943044, 136319052, 505086728, 1878395920, 7009239644, 26235435248, 98475145476, 370584275964, 1397918543552, 5284861554816

Offset: 1

Author

Jeppe Stig Nielsen, Jun 24 2000

Keywords

• nonn

Comments

Operation # can be interpreted as NOT AND. The ratio a(n)/A000108(n-1) converges to (2-sqrt(2))/2. Thanks to Soren Galatius Smith

Links

• G. C. Greubel, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A055113, A055392, A273958.

Programs

Mathematica

f[x_] := (1 - Sqrt[1 - 4*x])/2; CoefficientList[Series[(1 + 2*f[x] - Sqrt[1 + 4*(f[x])^2])/(2*x), {x, 0, 50}], x] (* G. C. Greubel, Jun 10 2016 *)

Formula

G.f.: 1 - (1/2)*(1 - 4*x)^(1/2) - (1/2)*(3 - 2*(1 - 4*x)^(1/2) - 4*x)^(1/2).

G.f.: (1 + 2*C(x) - sqrt(1 + 4*C(x)^2))/2, where C(x) = (1 - sqrt(1 - 4*x))/2 is the g.f. of the Catalan numbers (A000108). - Paul D. Hanna, Jun 10 2016

G.f. A(x) satisfies: A(x) = x + (A(x) - C(x))^2, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108). - Paul D. Hanna, Jun 11 2016