|
| |
|
|
A007858
|
|
G.f. is 1 - 1/f(x), where f(x) = 1+x+3*x^2+9*x^3+32*x^4+... is 1/x times g.f. for A063020.
|
|
2
|
|
|
|
1, 2, 4, 13, 44, 164, 636, 2559, 10556, 44440, 190112, 824135, 3612244, 15981632, 71277736, 320121747, 1446537564, 6571858168, 30000766128, 137544893940, 633051803120, 2923867281660, 13547594977500, 62955434735505, 293336372858724, 1370149533359784, 6414423856436816
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Number of maximal independent sets in rooted plane trees on n nodes. - Olivier Gérard, Jul 05 2001
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n+1) = Sum_{k = 1..n} ( binomial(n+k,k)/(n+k)*Sum_{j = 0..k} ( binomial(j,n-k-j+1)*binomial(k,j)*(-1)^(n+k-j+1) ) ) + C(n), where C(n) is a Catalan number. - Vladimir Kruchinin, Nov 13 2014
Recurrence: 16*(n-1)*n*(2*n-3)*(17*n^2 - 81*n + 96)*a(n) = (n-1)*(1819*n^4 - 14124*n^3 + 40377*n^2 - 50320*n + 23040)*a(n-1) + 8*(2*n-5)*(4*n-11)*(4*n-9)*(17*n^2 - 47*n + 32)*a(n-2). - Vaclav Kotesovec, Nov 14 2014
Asymptotics (Klazar, 1997): a(n) ~ sqrt(5731-4635/sqrt(17)) * ((107+51*sqrt(17))/64)^n / (256 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 14 2014
|
|
|
MAPLE
|
series(1-x/RootOf(_Z-_Z^2-_Z^3+_Z^4-x), x=0, 20); # Mark van Hoeij, May 28 2013
|
|
|
MATHEMATICA
|
Rest[CoefficientList[1-x/InverseSeries[Series[x-x^2-x^3+x^4, {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Nov 14 2014 *)
Table[Sum[Binomial[n + k, k]/(n + k)*Sum[(Binomial[j, n - k - j + 1]*
Binomial[k, j]*(-1)^(n + k - j + 1)), {j, 0, k}], {k, 1, n}] + CatalanNumber[n], {n, 0, 50}] (* G. C. Greubel, Feb 15 2017 *)
|
|
|
PROG
|
(PARI) x='x+O('x^66); Vec(1-x/serreverse(x-x^2-x^3+x^4)) \\ Joerg Arndt, May 28 2013
(Maxima)
a(n):=sum(binomial(n+k, k)/(n+k)*sum(binomial(j, n-k-j+1)*binomial(k, j)*(-1)^(n+k-j+1), j, 0, k), k, 1, n)+1/(n+1)*binomial(2*n, n); // Vladimir Kruchinin, Nov 13 2014
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
