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.
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
Offset: 1
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Martin Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
- A. Mironov and A. Morozov, Algebra of quantum C-polynomials, arXiv:2009.11641 [hep-th], 2020.
- Index entries for sequences related to rooted trees
Crossrefs
Cf. A000108.
Programs
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Maple
series(1-x/RootOf(Z-_Z^2-_Z^3+_Z^4-x), x=0,20); # _Mark van Hoeij, May 28 2013
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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 *) -
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 */
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PARI
my(x='x+O('x^66)); Vec(1-x/serreverse(x-x^2-x^3+x^4)) \\ Joerg Arndt, May 28 2013
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
Comments