Polynomial Sequences

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Problem

Given is a sequence of integers, which follow a specific polynomial progression. Your job is to determine the next number in the sequence.

Further information:
-The sequences will be given as a space separated list. The first term is n=1, the second term is n=2, etc.
-Individual sequences are separated by a new line.
-The polynomials used will be between the order of 2 and 6.
-Enough terms will be given to identify each sequence uniquely: order 2 will have at least 3 terms, order 3 will have at least 4 terms, etc.
-All polynomial coefficients will be integer.

Options

now post-mortem time, all source codes will be revealed

Sample input:_

1 4 9
1 8 27 64
1 16 81 256 625
1 32 243 1024 3125 7776
1 64 729 4096 15625 46656 117649

Sample output:

16
125
1296
16807
262144

Sample input:_

4 61 284 859 2044 4169
2 16 72 224 550
2 19 70 173 346
5 14 27 44 65 90
1 -22 -109 -326 -763
5 6 -1 -22
3 37 155 435 979
5 3 -9 -37
0 -10 -34 -78
6 66 300 900 2130 4326

Sample output:

7636
1152
607
119
-1534
-63
1913
-87
-148
7896

Sample input:_

9 113 717 2775 8033 19269 40533 77387
8 53 214 617 1436 2893
6 7 -26 -147 -434 -989 -1938
5 5 -3 -25
8 62 278 842 2012 4118 7562
8 20 20 -34 -208 -592 -1300 -2470
8 52 236 734 1792
3 27 -199 -2193 -10401 -34597 -93003 -216129 -451333
9 243 2485 13437 50241 148159 370413
5 18 45 92 165 270
9 35 91 189 341 559
10 130 830 3298 9802 24050 51550 99970 179498
5 6 -27 -148 -435 -990 -1939 -3432
6 83 640 2781 8666 21871 47748 93785
2 48 558 3296 13050 40032 102998
7 169 1475 7195 25039 70037 168139 360535 708695
8 29 84 191 368 633
4 30 96 220 420 714 1120
6 29 82 177
8 222 2376 13100 49440 146538 367472 815256 1647000

Sample output:

137145
5258
-3431
-67
12818
-4264
3728
-868101
820185
413
855
303202
-5643
169966
233088
1300129
1004
1656
326
3090230

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