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%I #29 Jan 02 2023 15:24:12
%S 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,
%T 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,
%U 119,60,20,5,1,0,1,12,78,219,350,325,202,83,25,5,1,0
%N Triangle of trees with n nodes and k leaves, 2 <= k <= n.
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 80, Problem 3.9.
%H Andrew Howroyd, <a href="/A055290/b055290.txt">Table of n, a(n) for n = 2..1226</a> (rows 2..50)
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F 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.
%e Triangle begins:
%e n=2: 1
%e n=3: 1 0
%e n=4: 1 1 0
%e n=5: 1 1 1 0
%e n=6: 1 2 2 1 0
%e n=7: 1 3 4 2 1 0
%e n=8: 1 4 8 6 3 1 0
%e n=9: 1 5 14 14 9 3 1 0
%e n=10: 1 7 23 32 26 12 4 1 0
%e n=11: 1 8 36 64 66 39 16 4 1 0
%e n=12: 1 10 54 123 158 119 60 20 5 1 0
%e n=13: 1 12 78 219 350 325 202 83 25 5 1 0
%o (PARI)
%o 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)}
%o 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))}
%o { my(A=T(10)); for(n=1, #A, print(A[n])) }
%Y Row sums give A000055, row sums with weight k give A003228.
%Y Columns 3 through 12: A001399(n-4), A055291, A055292, A055293, A055294, A055295, A055296, A055297, A055298, A055299.
%Y Cf. A055300, A055301, A238416, A304222.
%Y The labeled version is A055314.
%Y Central column is A358107.
%Y Left of central column is A359398.
%Y Cf. A000088, A000272, A001349, A001433, A002494, A185650.
%K nonn,tabl
%O 2,12
%A _Christian G. Bower_, May 09 2000
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