A030980 -id:A030980 - cp's OEIS frontend

A378079 Number of series-reduced noncrossing trees with n edges.

1, 1, 0, 4, 5, 33, 91, 408, 1485, 6195, 24838, 103752, 432796, 1834140, 7815900, 33591376, 145197017, 631281591, 2757917260, 12102728740, 53321334381, 235768155073, 1045889996047, 4653534540816, 20761857325000, 92862669150004, 416316199107096, 1870414803490240

Offset: 0

Andrew Howroyd, Nov 21 2024

nonn

			The a(3) = 4 trees are:
    o---o    o---o    o   o    o   o
    | \        / |    | /        \ |
    o   o    o   o    o---o    o---o

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000014, A001764, A030980, A378078.

seq(n)={my(g=serreverse(x/(1/(1-x)^2 - 2*x) + O(x*x^n))); Vec(1/(1 - g) - g^2)}

G.f.: 1/(1 - g(x)) - g(x)^2 where g(x) is the g.f. of A030980.

A030982 Number of noncrossing nonplanted bushes with n nodes, i.e., rooted noncrossing trees with n nodes and no nodes of degree 1.

Original entry on oeis.org

0, 1, 1, 7, 18, 80, 284, 1169, 4628, 19137, 79165, 333058, 1410608, 6029816, 25941384, 112315945, 488862888, 2138161043, 9391903131, 41414729419, 183264846010, 813564012660, 3622193670040, 16170171489820, 72364908958800, 324586284275500, 1458976377988636

Offset: 1

Emeric Deutsch

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to rooted trees

Cf. A030980, A030981.

Table[5*Sum[(-1)^(n-k)*2^(n-k)*k*Binomial[n,k]*Binomial[3*k-2,k-2]/ (2*k+1),{k,2,n}]/n,{n,1,20}] (* Vaclav Kotesovec, Oct 24 2012 *)

a(n) = 5*sum(k=2, n, (-1)^(n-k)*2^(n-k)*k*binomial(n,k)*binomial(3*k-2,k-2)/(2*k+1))/n; \\ Andrew Howroyd, Nov 12 2017

a(n) = 5*Sum_{k=2..n} ((-1)^(n-k)*2^(n-k)*k*C(n,k)*C(3*k-2,k-2)/(2*k+1))/n.
Recurrence: 2*n*(2*n+1)*a(n) = (n-1)*(11*n-12)*a(n-1) + 6*(9*n^2-21*n+8) * a(n-2) - 4*(n-3)*(11*n-56)*a(n-3) - 152*(n-4)*(n-3)*a(n-4). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ 5*19^(n+1/2)/(27*sqrt(Pi)*4^(n+1)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012
a(n) = A030981(n) - A030980(n). - Andrew Howroyd, Nov 12 2017

cp's OEIS Frontend

A378079 Number of series-reduced noncrossing trees with n edges.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula