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A055290 Triangle of trees with n nodes and k leaves, 2 <= k <= n. 18
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 1, 3, 4, 2, 1, 0, 1, 4, 8, 6, 3, 1, 0, 1, 5, 14, 14, 9, 3, 1, 0, 1, 7, 23, 32, 26, 12, 4, 1, 0, 1, 8, 36, 64, 66, 39, 16, 4, 1, 0, 1, 10, 54, 123, 158, 119, 60, 20, 5, 1, 0, 1, 12, 78, 219, 350, 325, 202, 83, 25, 5, 1, 0, 1, 14, 110, 377, 727 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,12

REFERENCES

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 80, Problem 3.9.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 2..1226 (rows 2..50)

Index entries for sequences related to trees

FORMULA

G.f.: A(x, y)=(1-x+x*y)*B(x, y)+(1/2)*(B(x^2, y^2)-B(x, y)^2), where B(x, y) is g.f. of A055277.

EXAMPLE

Triangle starts with:

n=2: 1;

n=3: 1,0;

n=4: 1,1,0;

n=5: 1,1,1,0;

n=6: 1,2,2,1,0;

...

PROG

(PARI)

EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i ))-1)}

T(n)={my(u=[y]); for(n=2, n, u=concat([y], EulerMT(u))); my(r=x*Ser(u), v=Vec(r*(1-x+x*y) + (substvec(r, [x, y], [x^2, y^2]) - r^2)/2)); vector(n-1, k, Vecrev(v[1+k]/y^2, k))}

{ my(A=T(10)); for(n=1, #A, print(A[n])) }

CROSSREFS

Row sums give A000055, row sums with weight k give A003228.

Columns 3 through 12: A001399(n-4), A055291, A055292, A055293, A055294, A055295, A055296, A055297, A055298, A055299.

Cf. A055300, A055301, A238416, A304222.

Sequence in context: A037836 A194522 A165013 * A125629 A339160 A141335

Adjacent sequences:  A055287 A055288 A055289 * A055291 A055292 A055293

KEYWORD

nonn,tabl

AUTHOR

Christian G. Bower, May 09 2000

STATUS

approved

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Last modified April 19 08:59 EDT 2021. Contains 343110 sequences. (Running on oeis4.)