sanity (Catalan, r=1, d=1): full rank = False (expected False);  (r=1, d=0): True (expected True)
m_2n                 terms=281  shapes (order r, degree <= d) with only the trivial solution: [(1, 135), (2, 89), (3, 66), (4, 52), (5, 43), (6, 37), (7, 32), (8, 28), (9, 25), (10, 22), (11, 20), (12, 19), (13, 17), (14, 16), (15, 15), (16, 14), (17, 13), (18, 12), (19, 11), (20, 11), (21, 10), (22, 9), (23, 9), (24, 8)]
     no candidate recurrence found
n! m_2n              terms=281  shapes (order r, degree <= d) with only the trivial solution: [(1, 135), (2, 89), (3, 66), (4, 52), (5, 43), (6, 37), (7, 32), (8, 28), (9, 25), (10, 22), (11, 20), (12, 19), (13, 17), (14, 16), (15, 15), (16, 14), (17, 13), (18, 12), (19, 11), (20, 11), (21, 10), (22, 9), (23, 9), (24, 8)]
     no candidate recurrence found
m_2n / n!            terms=281  shapes (order r, degree <= d) with only the trivial solution: [(1, 135), (2, 89), (3, 66), (4, 52), (5, 43), (6, 37), (7, 32), (8, 28), (9, 25), (10, 22), (11, 20), (12, 19), (13, 17), (14, 16), (15, 15), (16, 14), (17, 13), (18, 12), (19, 11), (20, 11), (21, 10), (22, 9), (23, 9), (24, 8)]
     no candidate recurrence found
m_2n / (2n)!         terms=281  shapes (order r, degree <= d) with only the trivial solution: [(1, 135), (2, 89), (3, 66), (4, 52), (5, 43), (6, 37), (7, 32), (8, 28), (9, 25), (10, 22), (11, 20), (12, 19), (13, 17), (14, 16), (15, 15), (16, 14), (17, 13), (18, 12), (19, 11), (20, 11), (21, 10), (22, 9), (23, 9), (24, 8)]
     no candidate recurrence found
k_2n (n >= 1)        terms=280  shapes (order r, degree <= d) with only the trivial solution: [(1, 134), (2, 89), (3, 66), (4, 52), (5, 43), (6, 37), (7, 32), (8, 28), (9, 25), (10, 22), (11, 20), (12, 19), (13, 17), (14, 16), (15, 15), (16, 14), (17, 13), (18, 12), (19, 11), (20, 11), (21, 10), (22, 9), (23, 9), (24, 8)]
     no candidate recurrence found
n! k_2n (n >= 1)     terms=280  shapes (order r, degree <= d) with only the trivial solution: [(1, 134), (2, 89), (3, 66), (4, 52), (5, 43), (6, 37), (7, 32), (8, 28), (9, 25), (10, 22), (11, 20), (12, 19), (13, 17), (14, 16), (15, 15), (16, 14), (17, 13), (18, 12), (19, 11), (20, 11), (21, 10), (22, 9), (23, 9), (24, 8)]
     no candidate recurrence found
k_2n / n! (n >= 1)   terms=280  shapes (order r, degree <= d) with only the trivial solution: [(1, 134), (2, 89), (3, 66), (4, 52), (5, 43), (6, 37), (7, 32), (8, 28), (9, 25), (10, 22), (11, 20), (12, 19), (13, 17), (14, 16), (15, 15), (16, 14), (17, 13), (18, 12), (19, 11), (20, 11), (21, 10), (22, 9), (23, 9), (24, 8)]
     no candidate recurrence found
nice -n 10 venv/bin/python  c_fast_N560.txt 560 24  3.81s user 0.03s system 99% cpu 3.844 total
