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A261000 Unordered even-degree bilabeled increasing trees on 2n+1 nodes.

%I A261000 #28 Jul 02 2024 21:46:01
%S A261000 1,3,189,68607,82908441,251944606683,1618221395188629,
%T A261000 19514714407120367127,405452689572115086887601,
%U A261000 13596354857453497541480646963,699110237190377161907394095173869,52888313306236766686682435536884784047
%N A261000 Unordered even-degree bilabeled increasing trees on 2n+1 nodes.
%H A261000 Markus Kuba and Alois Panholzer, <a href="http://arxiv.org/abs/1411.4587">Combinatorial families of multilabelled increasing trees and hook-length formulas</a>, arXiv:1411.4587 [math.CO], 2014. See p. 18
%F A261000 Kuba et al. (2014) gives a recurrence (see Theorem 7).
%F A261000 a(n) = A258659(2*n). - _Michael Somos_, Jun 17 2017
%p A261000 A261000aer := proc(n)
%p A261000     option remember;
%p A261000     local a,nloc,j,k,l;
%p A261000     if n = 1 then
%p A261000         1;
%p A261000     else
%p A261000         nloc := n-2 ;
%p A261000         a :=0 ;
%p A261000         for j from 0 to nloc-1 do
%p A261000             for k from 0 to nloc-1-j do
%p A261000                 l := nloc-1-j-k ;
%p A261000                 if l >= 0 then
%p A261000                     a := a+procname(j+1)*procname(k+1)*procname(l+1) * (2*nloc+1)!/(2*j+1)!/(2*k+1)!/(2*l+1)! ;
%p A261000                 end if;
%p A261000             end do:
%p A261000         end do:
%p A261000         %/2 ;
%p A261000     end if;
%p A261000 end proc:
%p A261000 A261000 := proc(n)
%p A261000     A261000aer(2*n+1) ;
%p A261000 end proc:
%p A261000 seq(A261000(n),n=0..15) ; # _R. J. Mathar_, Aug 18 2015
%t A261000 terms = 12; nmax = 4 terms; A = 1; Do[A = Exp[Integrate[A^(1/2)*Integrate[1/A^(3/2), x], x] + O[x]^nmax], nmax]; A258659 = CoefficientList[A, x^2]*Range[0, nmax - 2, 2]!;
%t A261000 a[n_] := A258659[[2 n + 1]];
%t A261000 Table[a[n], {n, 0, terms - 1}] (* _Jean-François Alcover_, Nov 27 2017 *)
%t A261000 a[ n_] := If[ n<0, 0, (-1)^n * (4*n+1)! * SeriesCoefficient[ JacobiSD[x, 1/2], {x, 0, 4*n+1}]]; (* _Michael Somos_, Sep 03 2022 *)
%t A261000 a[ n_] := If[ n<0, 0, (-1)^n * (4*n+1)! * SeriesCoefficient[ x*Sqrt[1/x^2 / WeierstrassP[x, {1, 0}]], {x, 0, 4*n+1}]]; (* _Michael Somos_, Jul 02 2024 *)
%t A261000 a[ n_] := If[ n<0, 0, (-1)^n * (4*n+1)! * SeriesCoefficient[
%t A261000 InverseSeries[ Series[ x * Hypergeometric2F1[1/4, 1/2, 5/4, x^4/4], {x, 0, 4*n+1}]], {x, 0, 4*n+1}]]; (* _Michael Somos_, Jul 02 2024 *)
%o A261000 (PARI) {a(n) = if( n<0, 0, my(m = 4*n + 1); m! * polcoeff( serreverse( intformal( 1 / sqrt(1 + x^4/4 + x * O(x^m)) ) ), m))}; /* _Michael Somos_, Jun 17 2017 */
%Y A261000 Closely related to A104203.
%Y A261000 Cf. A258659.
%K A261000 nonn
%O A261000 0,2
%A A261000 _N. J. A. Sloane_, Aug 09 2015