# OK to post homework
# Lucy Martinez, 04-03-22, Assignment 19
read `C19.txt`:
#----------------------Problem 1-----------------------#
# (i) Write a procedure, UmbT(P,var,N), that inputs a polynomial P in a list of variables, var, say 
# [x1,x2,x3, ..xk], and outputs the first N terms of the sequence that is the "umbral transform" under the umbra
# P^n under the umbra x1^a1 .... xk^ak -> (a1+...+ak)!/(a1!*...*ak!)
# (ii) What are the A-mumbers (if they exist) of the following sequences
# (a) UmbT(x+y,[x,y],20);
# (b) UmbT(x+y+z,[x,y,z],20);
# (c) UmbT(x1+x2+x3+x4,[x1,x2,x3,x4],20);
# (d) UmbT(1+2*x+3*y,[x,y],20);
# (e) UmbT((1+x)*(1+y)*(1+z),[x,y,z],20);


UmbT:=proc(P,var,N) local i,d:
[seq(Umb(P^i,var ),i=1..N)]:
end:

UmbT(x+y,[x,y],20);
# [2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 
#  10400600, 40116600, 155117520, 601080390, 2333606220, 9075135300, 35345263800, 137846528820]

UmbT(x+y+z,[x,y,z],20);
# [3, 15, 93, 639, 4653, 35169, 272835, 2157759, 17319837, 140668065, 1153462995, 
#  9533639025, 79326566595, 663835030335, 5582724468093, 47152425626559, 399769750195965, 
#  3400775573443089, 29016970072920387, 248256043372999089]


UmbT(x1+x2+x3+x4,[x1,x2,x3,x4],20);
# [4, 28, 256, 2716, 31504, 387136, 4951552, 65218204, 878536624, 
#  12046924528, 167595457792, 2359613230144, 33557651538688, 
#  481365424895488, 6956365106016256, 101181938814289564, 
#  1480129751586116848, 21761706991570726096, 321401321741959062016, 4766118425002290943216]


UmbT(1+2*x+3*y,[x,y],20);
# [6, 48, 432, 4104, 40176, 400896, 4053888, 41396832, 425922624, 4408432128, 45850157568,
#  478784058624, 5016552777216, 52713659768832, 555293062213632, 5862253175834112, 
#  62006462404125696, 656969858722013184, 6971258566157672448, 74074268834613817344]



UmbT((1+x)*(1+y)*(1+z),[x,y,z],20);
# [16, 550, 24136, 1176106, 60777016, 3261771136, 179738342800, 10099540746586,
#  576074600766736, 33252430325188900, 1938057963369265600, 113864747946771636400, 
#  6735066708925272520000, 400678521748749771325600, 23955811161103318460282656,
#  1438502844846334412424107866, 86710074762372112298341225600, 5244447235864729051389952560556, 
#  318158612150147855918832514776496, 19353875636742012377756474564054476]