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A061575 Number of planar planted trees with n non-root nodes and without isolated 2-valent nodes. 0
0, 1, 0, 2, 2, 7, 14, 41, 107, 307, 871, 2546, 7497, 22380, 67366, 204517, 625132, 1922700, 5945469, 18473841, 57649699, 180602285, 567772883, 1790663427, 5663969707, 17963483548, 57112388657, 181994536484, 581168157605 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

An isolated 2-valent node is a 2-valent node non-adjacent to any other 2-valent node.

REFERENCES

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.7.11).

LINKS

Table of n, a(n) for n=0..28.

FORMULA

G.f.: Sum_{i >= 0} 1/(i+1)*binomial(2*i, i)*x^(i+1)*((1+x^3)/(1-x^2))^(i+1)*(1+x*(1+x^3)/(1-x^2))^(-(i+1)) or (1-x^2+x^3-sqrt((1-x^2+x^3)*(1-4*x+3*x^2-3*x^3)))/(2-2*x^2+2*x^3).

a(n) = Sum_{k=1..n}(((Sum_{j=1..k}(binomial(2*j-2,j-1)*(-1)^(j-k)*binomial(k,j)))*Sum_{j=0..(n-k)/2}(binomial(k,j)*binomial(n-k-j-1,n-k-2*j)))/k). - Vladimir Kruchinin, Apr 15 2016

MATHEMATICA

Table[Sum[((Sum[Binomial[2 j - 2, j - 1] (-1)^(j - k) Binomial[k, j], {j, 1, k}]) Sum[Binomial[k, j] Binomial[n - k - j - 1, n - k - 2 j], {j,

0, (n - k)/2}])/k, {k, 1, n}], {n, 0, 28}] (* Michael De Vlieger, Apr 15 2016 *)

PROG

(Maxima)

a(n):=sum(((sum(binomial(2*j-2, j-1)*(-1)^(j-k)*binomial(k, j), j, 1, k))*sum(binomial(k, j)*binomial(n-k-j-1, n-k-2*j), j, 0, (n-k)/2))/k, k, 1, n); /* Vladimir Kruchinin, Apr 15 2016 */

CROSSREFS

Sequence in context: A162460 A187306 A061274 * A306009 A290646 A133602

Adjacent sequences:  A061572 A061573 A061574 * A061576 A061577 A061578

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Jun 13 2001

STATUS

approved

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Last modified April 23 01:23 EDT 2021. Contains 343198 sequences. (Running on oeis4.)