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 A060016 Triangle T(n,k) = number of partitions of n into k distinct parts, 1<=k<=n. 11
 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 1, 4, 3, 0, 0, 0, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 0, 0, 0, 1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0, 1, 5, 7, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 10, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,12 COMMENTS Also number of partitions of n-k(k+1)/2 into at most k parts (not necessarily distinct). A025147(n) = Sum(a(n-k+1,k-1): 1=0, z^n * q^((n^2+n)/2) / prod(k=1..n, 1-q^k ) ), rows by powers of q, columns by powers of z; includes row 0 (drop term for n=0 for this triangle, see PARI code); setting z=1 gives g.f. for A000009; cf. to g.f. for A072574. [Joerg Arndt, Oct 20 2012] EXAMPLE Triangle starts [ 1]  1, [ 2]  1, 0, [ 3]  1, 1, 0, [ 4]  1, 1, 0, 0, [ 5]  1, 2, 0, 0, 0, [ 6]  1, 2, 1, 0, 0, 0, [ 7]  1, 3, 1, 0, 0, 0, 0, [ 8]  1, 3, 2, 0, 0, 0, 0, 0, [ 9]  1, 4, 3, 0, 0, 0, 0, 0, 0, [10]  1, 4, 4, 1, 0, 0, 0, 0, 0, 0, [11]  1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0, [12]  1, 5, 7, 2, 0, 0, 0, 0, 0, 0, 0, 0, [13]  1, 6, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, [14]  1, 6, 10, 5, 0, 0, 0, 0, 0, 0, 0, 0, ... T(8,3)=2 since 8 can be written in 2 ways as the sum of 3 distinct positive integers: 5+2+1 and 4+3+1. MAPLE b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)       -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))     end: T:= proc(n) local l; l:= subsop(1=NULL, b(n, n));       l[], 0\$(n-nops(l))     end: seq(T(n), n=1..20);  # Alois P. Heinz, Dec 12 2012 MATHEMATICA Flatten[Table[Length[Select[IntegerPartitions[n, {k}], Max[Transpose[ Tally[#]][[2]]]==1&]], {n, 20}, {k, n}]] (* Harvey P. Dale, Feb 27 2012 *) T[_, 1] = 1; T[n_, k_] /; 1

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