#OK to post homework 
#Natalya Ter-Saakov, April 3, Assignment 19
read "C19.txt":

#Problem 1
#UmbT(P,var,N): 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
#x1a1 .... xkak -> (a1+...+ak)!/(a1!*...*ak!)
UmbT:=proc(P,var,N) local n:
[seq(Umb(P^n,var), n=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]
#A000984

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]
#A002893

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]
#A002895

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]
#Doesn't seem to exist in OEIS

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]
#Doesn't seem to exist