[seqfan] Finite sequence?

[jnthn stdhr]

Dear seqfans,

I have found a sequence related to partitions that appears to abruptly
halt.  Determining whether it is finite or not is beyond my mathematical
abilities.  I posted this question on math.stackexchange, but have yet to
get an answer:

Define T(n,k) = number of partitions of n in which the greatest part is k,
1≤k≤n (A008284). Let P be the normal partition function (A000041), and let
T_m denote the mth row of T(n,k).

Notice that whenever m=2P(n) for some integer n then T(m,2)=m/2 and for
some other 2<k<m−2, T(m,k) also equals m/2. For example, let m=2P(4)=10,
and note T(10,2)=T(10,6)=5. A similar result is true for m=2P(n)+1; for
example with m=2P(3)+1, T(7,2)=T(7,4)=3. So for every m=2P(n), there will
be repeated values greater than 1 in the sequences T_m and T_m+1.

For the values of m=8,13,19,26,34,43,46,68, none of which is equal to 2P(n)
or 2P(n)+1 for some integer n, there are repeated values greater than 1 in
T_m:

8: [1, 4, (5), (5), 3, 2, 1, 1]

13: [1, 6, 14, (18), (18), 14, 11, 7, 5, 3, 2, 1, 1]

19: [1, 9, (30), 54, 70, 71, 65, 52, 41, (30), 22, 15, 11, 7, 5, 3, 2, 1, 1]

26: [1, 13, (56), 136, 221, 282, 300, 288, 252, 212, 169, 133, 101, 77,
(56), 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]

34: [1, 17, 96, (297), 603, 931, 1175, 1297, 1291, 1204, 1060, 905, 747,
608, 483, 383, (297), 231, 176, 135, 101, 77, 56, 42, 30, 22, 15, 11, 7, 5,
3, 2, 1, 1]

43: [1, 21, 154, 588, 1469, 2702, 4011, 5066, (5708), 5888, (5708), 5262,
4691, 4057, 3446, 2871, 2369, 1928, 1563, 1251, 1001, 792, 627, 490, 385,
297, 231, 176, 135, 101, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]

46: [1, 23, (176), 720, 1898, 3692, 5731, 7564, 8824, 9418, 9373, 8877,
8073, 7139, 6158, 5231, 4370, 3621, 2965, 2417, 1951, 1573, 1255, 1002,
792, 627, 490, 385, 297, 231, (176), 135, 101, 77, 56, 42, 30, 22, 15, 11,
7, 5, 3, 2, 1, 1]

68: [1, 34, (385), 2280, 8529, 22856, 47527, 81457, 120092, 157564, 188556,
210005, 220877, 222118, 215415, 203067, 187013, 169120, 150614, 132559,
115463, 99795, 85628, 73088, 62049, 52490, 44210, 37143, 31088, 25970,
21618, 17970, 14881, 12310, 10143, 8349, 6842, 5604, 4565, 3718, 3010,
2436, 1958, 1575, 1255, 1002, 792, 627, 490, (385), 297, 231, 176, 135,
101, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]

I have checked up to m=5,000 and found no additional rows T_m with m>68
that have repeated values greater than 1, except when m can be written as
2P(n) or 2P(n)+1 for some integer n.

Is it possible that no more sporadic examples exist?

As an aside, would some mind timing how long it takes Mathematica to
produce the triangle A008284(10,000)?

My email access is intermittent, so my replys may be delayed.

-Jonathan