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A261002
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Number of unordered unilabeled-bilabeled increasing trees.
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2
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1, 2, 4, 14, 66, 392, 2806, 23490, 225104, 2429538, 29159162, 385196964, 5553689194, 86776290538, 1460602599748, 26346881876166, 507037223792778, 10369313541925896, 224565589119822926, 5134144053785703610, 123569877999804177120, 3123085821618415219082
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OFFSET
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1,2
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LINKS
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FORMULA
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Kuba et al. (2014) give a recurrence (see Th. 12).
E.g.f. A(x) satisfies A(x) = log(B'(x)) where B() is the e.g.f. of A261001. - Michael Somos, Jan 17 2017
E.g.f. A(x) satisfies A''' = A'' * (A' + A'' / (1 + A')). - Michael Somos, Jan 18 2017
a(n) ~ d^n * (n-1)!, where d = 1.20456083370247231... - Vaclav Kotesovec, Aug 17 2018
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EXAMPLE
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G.f. = x + 2*x^2 + 4*x^3 + 14*x^4 + 66*x^5 + 392*x^6 + 2806*x^7 + ...
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MATHEMATICA
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terms = 22;
A[_] = 0;
Do[A[x_] = Integrate[1 + Integrate[A'[x] (A'[x] + A[x]), x], x] + O[x]^(terms+2), {terms+2}];
nmax = 25; Q[0] = 0; Q[1] = 1; Do[Q[m] = Sum[Binomial[m - 2, k]*(Q[k] + Q[k + 1])*Q[m - k - 1], {k, 0, m - 2}], {m, 2, nmax}]; Table[Q[n] + Q[n-1], {n, 1, nmax}] (* Vaclav Kotesovec, Aug 17 2018 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = O(x); for(k=0, n, A = intformal( 1 + intformal( A' * (A' + A)))); n! * polcoeff( log(A'), n))}; /* Michael Somos, Jan 18 2017 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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