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A178519 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and having k vertices of outdegree 2 that have (two) leaves as their (two) children. 1
1, 1, 1, 1, 4, 1, 11, 3, 32, 10, 98, 33, 1, 309, 114, 6, 998, 402, 30, 3285, 1439, 137, 1, 10981, 5205, 600, 10, 37178, 18976, 2562, 70, 127227, 69610, 10758, 416, 1, 439369, 256626, 44640, 2250, 15, 1529280, 949974, 183594, 11452, 140, 5359314, 3528725 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row n has 1 + floor(n/3) entries (n >= 3).

Sum of entries in row n is A000108(n) (Catalan numbers).

T(n,0) = A178520(n).

Sum_{k>=0} k*T(n,k) = binomial(2n-5, n-2) = A001700(n-3).

Statistic suggested by Lou Shapiro.

LINKS

Table of n, a(n) for n=0..47.

FORMULA

G.f. G=G(t,z) satisfies z*G^2 - (1 - z^3 + tz^3)*G + 1 - z^2 + tz^2 = 0.

EXAMPLE

T(2,1)=1 because we have \/ .

Triangle starts:

   1;

   1;

   1,  1;

   4,  1;

  11,  3;

  32, 10;

  98, 33,  1;

MAPLE

eq := z*G^2-(1-z^3+t*z^3)*G+1-z^2+t*z^2: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; 1, 1; for n from 3 to 15 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form

CROSSREFS

Cf. A000108, A178520, A001700.

Sequence in context: A111964 A242351 A124324 * A094503 A113897 A158753

Adjacent sequences:  A178516 A178517 A178518 * A178520 A178521 A178522

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, May 31 2010

STATUS

approved

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Last modified July 29 09:41 EDT 2021. Contains 346344 sequences. (Running on oeis4.)