A222006 -id:A222006 - cp's OEIS frontend

A088327 G.f.: exp(Sum_{k>=1} B(x^k)/k), where B(x) = x + 2x^2 + 5x^3 + 14x^4 + 42x^5 + ... = (C(x)-1)/x and C is the g.f. for the Catalan numbers A000108.

1, 1, 3, 8, 25, 77, 256, 854, 2940, 10229, 36124, 128745, 463137, 1677816, 6118165, 22432778, 82660369, 305916561, 1136621136, 4238006039, 15852603939, 59471304434, 223704813807, 843547443903, 3188064830876, 12074092672950, 45816941923597, 174173975322767

Offset: 0

N. J. A. Sloane, Nov 06 2003

nonn

a(n) is the number of forests of rooted plane binary trees (each node has outdegree = 0 or 2) where the trees have a total of n internal nodes. Cf. A222006. - Geoffrey Critzer, Feb 26 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Row sums of A275431.

Cf. A222006, A304787.

m:=35;
f:= func< x | (&*[1/(1-x^j)^Catalan(j): j in [1..m+2]]) >;
R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( f(x) )); // G. C. Greubel, Dec 12 2022

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
       binomial(2*d, d)/(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 10 2012

With[{nn=35}, CoefficientList[Series[Product[1/(1-x^i)^CatalanNumber[i], {i,nn}], {x,0,nn}], x]] (* Geoffrey Critzer, Feb 26 2013 *)

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(2*n, n)/(n+1))
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

Euler transform of Catalan numbers (A000108). - Franklin T. Adams-Watters, Mar 01 2006

a(n) ~ c * 4^n / n^(3/2), where c = exp(Sum_{k>=1} (-2 + 4^k - 4^k*sqrt(1 - 4^(1-k)))/(2*k) ) / sqrt(Pi) = 1.60022306097485382475864802335610662545... - Vaclav Kotesovec, Mar 21 2021

A342770 T(n,k) is the number of rooted plane binary forests with n nodes and k trees: triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 3, 0, 1, 0, 1, 0, 5, 0, 3, 0, 1, 0, 1, 0, 0, 7, 0, 3, 0, 1, 0, 1, 0, 14, 0, 8, 0, 3, 0, 1, 0, 1, 0, 0, 22, 0, 8, 0, 3, 0, 1, 0, 1, 0, 42, 0, 24, 0, 8, 0, 3, 0, 1, 0, 1, 0, 0, 66, 0, 25, 0, 8, 0

Offset: 0

R. J. Mathar, Mar 21 2021

Multiset transform of A126120.

			See A222006 showing T(6,k).
The triangle starts (n>=0, 0<=k<=n):
  1
  0   1
  0   0   1
  0   1   0   1
  0   0   1   0   1
  0   2   0   1   0   1
  0   0   3   0   1   0   1
  0   5   0   3   0   1   0   1
  0   0   7   0   3   0   1   0   1
  0  14   0   8   0   3   0   1   0   1
  0   0  22   0   8   0   3   0   1   0   1
  0  42   0  24   0   8   0   3   0   1   0   1
  0   0  66   0  25   0   8   0   3   0   1   0   1
  0 132   0  74   0  25   0   8   0   3   0   1   0   1
  0   0 217   0  76   0  25   0   8   0   3   0   1   0   1

Cf. A222006 (row sums), A126120 (column k=1), A007595 (k=2), A046342 (k=3), A088327 (limit n->oo, row reverse).

cp's OEIS Frontend

A088327 G.f.: exp(Sum_{k>=1} B(x^k)/k), where B(x) = x + 2x^2 + 5x^3 + 14x^4 + 42x^5 + ... = (C(x)-1)/x and C is the g.f. for the Catalan numbers A000108.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A342770 T(n,k) is the number of rooted plane binary forests with n nodes and k trees: triangle read by rows.

Views

Author

Keywords

Comments

Examples

Crossrefs

cp's OEIS Frontend

A088327 G.f.: exp(Sum_{k>=1} B(x^k)/k), where B(x) = x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ... = (C(x)-1)/x and C is the g.f. for the Catalan numbers A000108.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A342770 T(n,k) is the number of rooted plane binary forests with n nodes and k trees: triangle read by rows.

Views

Author

Keywords

Comments

Examples

Crossrefs

A088327 G.f.: exp(Sum_{k>=1} B(x^k)/k), where B(x) = x + 2x^2 + 5x^3 + 14x^4 + 42x^5 + ... = (C(x)-1)/x and C is the g.f. for the Catalan numbers A000108.