cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A089436 Number of non-crossing connected graphs on n nodes on a circle in which a fixed (distinguished) node has degree one.

Original entry on oeis.org

1, 2, 9, 54, 374, 2820, 22485, 186494, 1592778, 13914108, 123750874, 1116809628, 10201516332, 94140605832, 876332565837, 8219124900558, 77594375595266, 736785675010380, 7031930543228910, 67420537625021460, 649070964647075700
Offset: 2

Views

Author

Emeric Deutsch, Dec 28 2003

Keywords

Comments

Convolution of (1, A007297) with itself.

Examples

			a(3)=2 because among the four non-crossing graphs on the points A,B,C, the distinguished node A has degree equal to 1 only in the graphs {AB,BC} and {AC,BC}; in the other two graphs ({AB,AC} and {AB,BC,AC}) the node A has degree 2.

Links

Crossrefs

Column k=1 of A143022.
Cf. A007297.

Programs

  • Mathematica
    terms = 21;
g[x_] = 0;
Do[g[x_] = g[x]^2 + x (1 + g[x])^3 + O[x]^(terms+2), {terms+2}];
Drop[CoefficientList[(x + x g[x])^2 + O[x]^(terms+2), x], 2] (* Jean-François Alcover, Oct 05 2011, updated Jul 29 2018 after Andrew Howroyd *)
  • PARI
    a(n)=if(n<3, n==2, sum(k=2, 2*n-4, 2*binomial(k-2, n-3)*binomial(3*n-5, 2*n-k-4))/(n-2)); \\ Andrew Howroyd, Nov 12 2017
  • PARI
    Vec((x+x*serreverse((x-x^2)/(1+x)^3 + O(x^25)))^2) \\ Andrew Howroyd, Nov 13 2017

Formula

a(n) = Sum_{k=2..2*n-4} 2*binomial(k-2, n-3)*binomial(3*n-5, 2*n-k-4)/(n-2) for n > 2. - Andrew Howroyd, Nov 12 2017
G.f.: g^2, where g satisfies g^3+g^2-3zg+2z^2=0, g(0)=0, or, in Maple notation, g := -1/3+(2/3)*sqrt(1+9*z)*sin((1/3)*arcsin((2+27*z+54*z^2)/2/(1+9*z)^(3/2))).
G.f.: (x+x*g)^2 where g satisfies g - g^2 = x*(1 + g)^3. - Andrew Howroyd, Nov 13 2017
a(n) ~ 2^(n-1) * 3^(3*n/2-9/4) / (sqrt(Pi)*n^(3/2)*sqrt(45+26*sqrt(3))). - Vaclav Kotesovec, Mar 17 2014
D-finite with recurrence n*(2*n-3)*(n-2)*a(n) +6*(9*n-10)*a(n-1) -12*(3*n-10)*(3*n-8)*(2*n-1)*a(n-2)=0. - R. J. Mathar, May 10 2018

A143023 Sum of the root degrees over all non-crossing connected graphs on n nodes on a circle (by root we mean a distinguished node).

Original entry on oeis.org

1, 6, 41, 306, 2422, 19980, 169941, 1479786, 13127114, 118217268, 1077955034, 9932655348, 92342765868, 865126386072, 8159523358029, 77411610053658, 738263424935170, 7073522484902820, 68056887469098990, 657269559836605980
Offset: 2

Views

Author

Emeric Deutsch, Jul 30 2008

Keywords

Comments

The number of non-crossing connected graphs on n nodes on a circle is given in A007297.

Examples

			a(3)=6 because in the graphs (AB,BC,CA), (AB,AC), (AB,BC) and (AC,BC) the root, say A, has degrees 2, 2, 1 and 1, respectively.

Links

Crossrefs

Cf. A143022, A007297.

Programs

  • Maple
    eq:=G*(1-G)-z*(1+G)^3: G:=RootOf(eq,G): Gser:=series(z*G*(1+G)/(1-G),z=0,25): seq(coeff(Gser,z,n),n=2..21); # end
L:=proc(p,q,r) options operator, arrow: sum(binomial(q,i)*binomial(r+p-1-i,r-1),i=0..min(p,q)) end proc: a:=proc(n) options operator, arrow: (L(n-2,3*n-2, n+1)+L(n-3,3*n-3,n))/(n-1) end proc: seq(a(n),n=2..21);
  • Mathematica
    L[p_, q_, r_] := Sum[ Binomial[q, i]*Binomial[r + p - 1 - i, r-1], {i, 0, Min[p, q]}]; t[n_, k_] := 2^(k-1)*(k*L[n - k - 1, 3*n - k - 4, n - 2] - L[n - k - 2, 3*n - k - 3, n-1])/(n-1); t[2, 1] = 1; a[n_] := Sum[ k*t[n, k], {k, 1, n-1}]; Table[a[n], {n, 2, 21}] (* Jean-François Alcover, Oct 05 2011, after Maple *)
  • PARI
    {my(n=25); my(g=serreverse(x*(1-x)/(1+x)^3 + O(x*x^n))); Vec(g*(1+g)/(1-g))} \\ Andrew Howroyd, Dec 22 2017

Formula

a(n) = (L(n-2, 3n-2, n+1) + L(n-3, 3n-3, n))/(n-1) where L(p,q,r)=[u^p](1+u)^q/(1-u)^r = Sum_{i=0..min(p,q)} binomial(q,i) * binomial(r+p-1-i, r-1).
G.f.: zG(1+G)/(1-G) where G=G(z) satisfies G(1-G)=z(1+G)^3.
a(n) = Sum_{k=1..n-1} k*A143022(n,k).
D-finite with recurrence 5*n*(n-1)*a(n) -18*(n-1)*(n-3)*a(n-1) +12*(-45*n^2+180*n-178)*a(n-2) +216*(3*n-10)*(3*n-11)*a(n-3)=0. - R. J. Mathar, Jul 26 2022
Showing 1-2 of 2 results.

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