A199086 T(n,k) = Number of partitions of n+2k-2 into parts >= k.
1, 1, 2, 1, 2, 3, 1, 2, 2, 5, 1, 2, 2, 4, 7, 1, 2, 2, 3, 4, 11, 1, 2, 2, 3, 4, 7, 15, 1, 2, 2, 3, 3, 5, 8, 22, 1, 2, 2, 3, 3, 5, 6, 12, 30, 1, 2, 2, 3, 3, 4, 5, 9, 14, 42, 1, 2, 2, 3, 3, 4, 5, 7, 10, 21, 56, 1, 2, 2, 3, 3, 4, 4, 6, 8, 13, 24, 77, 1, 2, 2, 3, 3, 4, 4, 6, 7, 11, 17, 34, 101, 1, 2, 2, 3, 3, 4, 4, 5
Offset: 1
Examples
All solutions for n=5, k=3: 3+3+3, 3+6, 4+5, 9.
Links
- R. H. Hardin and Alois P. Heinz, Table of n, a(n) for n = 1..10011
Crossrefs
Programs
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Maple
b:= proc(n, i) option remember; if n<0 then 0 elif n=0 then 1 elif i>n then 0 else b(n-i, i) +b(n, i+1) fi end: T:= (n, k)-> b(n+2*k-2, k): seq(seq(T(n, d+1-n), n=1..d), d=1..20); # Alois P. Heinz, Nov 06 2011 -
Mathematica
b[n_, i_] := b[n, i] = Which[n < 0, 0, n == 0, 1, i > n, 0, True, b[n - i, i] + b[n, i + 1]]; T[n_, k_] := b[n + 2*k - 2, k]; Table[Table[T[n, d + 1 - n], {n, 1, d}], {d, 1, 20}] // Flatten (* Jean-François Alcover, Jan 23 2016, after Alois P. Heinz *)
Formula
G.f. of column k: x^(2-2*k) * Product_{j>=k} 1/(1-x^j). - Alois P. Heinz, Nov 06 2011
Comments