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 A116598 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 1 (n>=0, 0<=k<=n). 5
 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 4, 2, 2, 1, 1, 0, 1, 4, 4, 2, 2, 1, 1, 0, 1, 7, 4, 4, 2, 2, 1, 1, 0, 1, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 21, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 24, 21, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,11 COMMENTS Row sums yield the partition numbers (A000041). Reversed rows converge to A002865. - Joerg Arndt, Jul 07 2014 T(n,k) is the number of partitions of n for which the difference between the two largest, not necessarily distinct, parts is k (in partitions having only 1 part, we assume that 0 is also a part). This follows easily from the definition by taking the conjugate partitions. Example: T(6,2) = 2 because we have [3,1,1,1] and [4,2]. - Emeric Deutsch, Dec 05 2015 LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA G.f.: G(t,x) = 1/( (1-t*x)*prod(j>=2, 1-x^j ) ). T(n,k) = p(n-k)-p(n-k-1) for k=1. Column k has g.f. x^k/prod(j>=2, 1-x^j ) (k>=0). EXAMPLE T(6,2) = 2 because we have [4,1,1] and [2,2,1,1]. Triangle starts: 00:   1, 01:   0,  1, 02:   1,  0,  1, 03:   1,  1,  0,  1, 04:   2,  1,  1,  0,  1, 05:   2,  2,  1,  1,  0,  1, 06:   4,  2,  2,  1,  1,  0,  1, 07:   4,  4,  2,  2,  1,  1,  0,  1, 08:   7,  4,  4,  2,  2,  1,  1,  0,  1, 09:   8,  7,  4,  4,  2,  2,  1,  1,  0,  1, 10:  12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1, 11:  14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1, 12:  21, 14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1, 13:  24, 21, 14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1, 14:  34, 24, 21, 14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1, 15:  41, 34, 24, 21, 14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1, ... MAPLE with(combinat): T:=proc(n, k) if k

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Last modified October 20 10:00 EDT 2019. Contains 328257 sequences. (Running on oeis4.)