#OK to post homework 
#Zhizhang Deng, 12/12/2020, Assignment 22

1.
    (i).
        L := TreeSeqL(67, t);
        coeff(L[67], t, 12);
        7626680355957910082617689180505462476268681313077891549163001398\

        47357281766830345010754357481701376000000000000000

    (ii). 
        ATreeSeq(40, 4)[40] = 90324445150366623501655158316607196285055246080819484164096000000000

2. 
    SeqRTchild := proc(S, N) local egf, z, s, L, n: 
        egf := 1;

        for s in S do: 
            egf := egf + z^s/s!
        end do; 

        L := FunEqToSeq(egf, z, N); 
        
        return [seq(L[n] * n!, n = 1..N)];
    end:
    (i). 
        SeqRTchild({1, 2}, 30) in OEIS, A36774
    (ii). 
        SeqRTchild({1, 2, 3}, 30) in OEIS, A36775
    (iii). 
        SeqRTchild({1, 2, 3, 4}, 30) in OEIS, A36776
    (iv). 
        SeqRTchild({1, 3, 4}, 30) not in OEIS
    (v). 
        SeqRTchild({2, 3, 4}, 30) not in OEIS

3. 
    SeqRTchildNone := proc(S, N) local egf_modifier, z, s, L, n: 
        egf_modifier := 0;

        for s in S do: 
            egf_modifier := egf_modifier + z^s/s!
        end do; 

        L := FunEqToSeq(exp(z) - egf_modifier, z, N); 
        
        return [seq(L[n] * n!, n = 1..N)];
    end:

    All of the S are not in OEIS. 

4. 
    (i).
        L := TreeSeqL(50, t);
        seq((AveAndMoms(L[n], t, 6)/n)[1], n = 3 .. 50); = 0.4444444443, 0.4218750000, 0.4096000000, 0.4018775720, 0.3965694566, 0.3926959038, 0.3897443431, 0.3874204890, 0.3855432895, 0.3839952306, 0.3826967067, 0.3815918726, 0.3806403927, 0.3798124058, 0.3790853319, 0.3784417801, 0.3778681390, 0.3773536026, 0.3768894829, 0.3764687155, 0.3760854967, 0.3757350149, 0.3754132467, 0.3751168023, 0.3748428052, 0.3745887982, 0.3743526714, 0.3741326000, 0.3739270013, 0.3737344922, 0.3735538615, 0.3733840426, 0.3732240931, 0.3730731769, 0.3729305497, 0.3727955466, 0.3726675726, 0.3725460922, 0.3724306237, 0.3723207314, 0.3722160212, 0.3721161345, 0.3720207460, 0.3719295583, 0.3718422998, 0.3717587219, 0.3716785965, 0.3716017144. 
        Seems like the limit is around 0.3716

    (ii). 
        For 
            2nd moment: ~ 0.43999
            3rd moment: ~ 0.00036
            4th moment: ~ 0.060
            5th moment: ~ 0.036
            6th moment: ~ 0.2934