|
| |
|
|
A202839
|
|
Number of stacks of length 1 in all 2ndary structures of size n.
|
|
6
|
|
|
|
0, 0, 0, 1, 3, 6, 16, 43, 110, 284, 733, 1886, 4853, 12486, 32121, 82647, 212699, 547552, 1410023, 3632260, 9360140, 24129284, 62224692, 160522287, 414246823, 1069376386, 2761502201, 7133442743, 18432633823, 47643696626, 123182434292, 318575889057, 824125660356
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
Number of stacks of length 2 in all 2ndary structures of size n+2.
Number of stacks of length 3 in all 2ndary structures of size n+4.
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum(k*A202841(n+2,k), k>=0).
a(n) = Sum(k*A202843(n+4,k), k>=0).
G.f.: g(z) = z^2*(1-z^2)^2*S(S - 1)/(1 - z + z^2 -2*z^2*S), where S is defined by S = 1 + z*S + z^2*S(S-1) (the g.f. of the secondary structure numbers A004148).
a(n) ~ 5^(3/4) * phi^(2*n-3) / (2*sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 29 2022
D-finite with recurrence -(n+2)*(406*n-3981)*a(n) +(2022*n^2-15917*n-13552)*a(n-1) +4*(-402*n^2+2594*n+593)*a(n-2) +4*(-605*n^2+7719*n-23415)*a(n-3) +4*(-203*n^2-527*n+15295)*a(n-4) +2*(804*n^2-8404*n+14555)*a(n-5) +(2826*n^2-42913*n+153174)*a(n-6) -(1210*n-6753)*(n-10)*a(n-7)=0. - R. J. Mathar, Jul 26 2022
|
|
|
EXAMPLE
|
a(5)=6: representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv, ABvBA; they have 0,1,1,1,1,1,1,0 stacks of length 1, respectively.
|
|
|
MAPLE
|
g := z^2*(1-z^2)^2*S*(S-1)/(1-z+z^2-2*z^2*S): S := ((1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^2: gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);
|
|
|
MATHEMATICA
|
CoefficientList[Series[-(1 - x^2)^2 * ((1 - x) + (-1 + 2*x + x^3) / Sqrt[(1 - 3*x + x^2) * (1 + x + x^2)]) / (2*x^2), {x, 0, 40}], x] (* Vaclav Kotesovec, May 29 2022 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
