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Number of label-increasing forests with branching bounded by 4.
7

%I #14 Oct 26 2019 09:59:05

%S 1,1,1,2,6,24,119,703,4819,37596,328871,3187627,33907426,392693219,

%T 4918535904,66246158633,954739609059,14660169150206,238933675330056,

%U 4119463404086442,74907534899129351,1432718565376182929

%N Number of label-increasing forests with branching bounded by 4.

%C See Riordan 1978 or 1979 for precise definition.

%H John Riordan, <a href="/A297196/a297196.pdf">Forests of label-increasing trees</a>, annotated scanned copy of 1978 pre-publication version.

%H John Riordan, <a href="https://doi.org/10.1002/jgt.3190030204">Forests of label-increasing trees</a>, J. Graph Theory, 3 (1979), 127-133.

%F E.g.f. F(x) satisfies the ODE: F'(x) = Sum_{j=0..4} (F(x)-1)^j/j! with F(0)=1. - _Max Alekseyev_, Jul 12 2019

%p Order := 25; F := rhs( dsolve( { diff(y(x),x) = sum((y(x)-1)^j/j!,j=0..4), y(0)=1 }, y(x), type=series ) ); seq( coeff(F,x,n)*n!, n=0..24 ); # _Max Alekseyev_, Jul 12 2019

%t m = 22; F[_] = 0;

%t Do[F[x_] = 1 + Integrate[Sum[(F[x] - 1)^j/j!, {j, 0, 4}], x] + O[x]^m // Normal, {m}];

%t CoefficientList[F[x], x]*Range[0, m - 1]! (* _Jean-François Alcover_, Oct 26 2019 *)

%Y Cf. A297196, A297197, A297198, A297201, A297202, A297203, A297204.

%K nonn

%O 0,4

%A _N. J. A. Sloane_, Jan 10 2018

%E Edited and more terms added by _Max Alekseyev_, Jul 12 2019