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

1, 3, 189, 68607, 82908441, 251944606683, 1618221395188629, 19514714407120367127, 405452689572115086887601, 13596354857453497541480646963, 699110237190377161907394095173869, 52888313306236766686682435536884784047

Offset: 0

Author

N. J. A. Sloane, Aug 09 2015

Keywords

• nonn

Links

• Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014. See p. 18

Crossrefs

Closely related to A104203.

Cf. A258659.

Programs

Maple

A261000aer := proc(n)
    option remember;
    local a,nloc,j,k,l;
    if n = 1 then
        1;
    else
        nloc := n-2 ;
        a :=0 ;
        for j from 0 to nloc-1 do
            for k from 0 to nloc-1-j do
                l := nloc-1-j-k ;
                if l >= 0 then
                    a := a+procname(j+1)*procname(k+1)*procname(l+1) * (2*nloc+1)!/(2*j+1)!/(2*k+1)!/(2*l+1)! ;
                end if;
            end do:
        end do:
        %/2 ;
    end if;
end proc:
A261000 := proc(n)
    A261000aer(2*n+1) ;
end proc:
seq(A261000(n),n=0..15) ; # R. J. Mathar, Aug 18 2015

Mathematica

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]!;
a[n_] := A258659[[2 n + 1]];
Table[a[n], {n, 0, terms - 1}] (* Jean-François Alcover, Nov 27 2017 *)
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 *)
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 *)
a[ n_] := If[ n<0, 0, (-1)^n * (4*n+1)! * SeriesCoefficient[
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 *)

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 */

Formula

Kuba et al. (2014) gives a recurrence (see Theorem 7).

a(n) = A258659(2*n). - Michael Somos, Jun 17 2017