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A261001 Numbers Q_n arising in computation of number of unordered unilabelled-bilabelled increasing trees. 2
1, 1, 3, 11, 55, 337, 2469, 21021, 204083, 2225455, 26933707, 358263257, 5195425937, 81580864601, 1379021735147, 24967860141019, 482069363651759, 9887244178274137, 214678344941548789, 4919465708844154821, 118650412290960022299, 3004435409327455196783 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..435 (terms 1..100 from Lars Blomberg)

Markus Kuba, Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014.

FORMULA

Kuba et al. (2014) give a recurrence (see Th. 12).

E.g.f. A(x) satisfies: A'' = A' * (A' + A). - Michael Somos, Jan 17 2017

a(n) + a(n-1) = A261002(n). - Michael Somos, Jan 17 2017

a(n) ~ d^n * (n-1)!, where d = 1.20456083370247231... - Vaclav Kotesovec, Aug 17 2018

EXAMPLE

G.f. = x + x^2 + 3*x^3 + 11*x^4 + 55*x^5 + 337*x^6 + 2469*x^7 + ...

MATHEMATICA

terms = 22;

A[_] = 0;

Do[A[x_] = Integrate[1+Integrate[A'[x] (A'[x]+A[x]), x], x] + O[x]^(terms+1) // Normal, {terms+1}];

Rest[CoefficientList[A[x], x]] Range[terms]! (* Jean-Fran├žois Alcover, Aug 16 2018, after Michael Somos *)

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], {n, 1, nmax}] (* Vaclav Kotesovec, Aug 17 2018 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = O(x); for(k=1, n, A = intformal( 1 + intformal( A' * (A' + A)))); n! * polcoeff(A, n))}; /* Michael Somos, Jan 18 2017 */

CROSSREFS

Cf. A261002.

Sequence in context: A330041 A125696 A001776 * A207556 A180875 A136104

Adjacent sequences:  A260998 A260999 A261000 * A261002 A261003 A261004

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Aug 09 2015

EXTENSIONS

More terms from Lars Blomberg, Aug 20 2015

STATUS

approved

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Last modified December 13 06:26 EST 2019. Contains 329968 sequences. (Running on oeis4.)