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A199086
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T(n,k) = Number of partitions of n+2k-2 into parts >= k.
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1
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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
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OFFSET
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1,3
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COMMENTS
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Row n goes to floor(n/2)+1
Table starts
...1...1...1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1
...2...2...2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2
...3...2...2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2
...5...4...3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3
...7...4...4..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3
..11...7...5..5..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4
..15...8...6..5..5..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4
..22..12...9..7..6..6..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5
..30..14..10..8..7..6..6..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5
..42..21..13.11..9..8..7..7..6..6..6..6..6..6..6..6..6..6..6..6..6..6..6..6..6
..56..24..17.12.10..9..8..7..7..6..6..6..6..6..6..6..6..6..6..6..6..6..6..6..6
..77..34..21.16.13.11.10..9..8..8..7..7..7..7..7..7..7..7..7..7..7..7..7..7..7
.101..41..25.18.15.12.11.10..9..8..8..7..7..7..7..7..7..7..7..7..7..7..7..7..7
.135..55..33.24.18.16.13.12.11.10..9..9..8..8..8..8..8..8..8..8..8..8..8..8..8
.176..66..39.27.21.17.15.13.12.11.10..9..9..8..8..8..8..8..8..8..8..8..8..8..8
.231..88..49.34.26.21.18.16.14.13.12.11.10.10..9..9..9..9..9..9..9..9..9..9..9
.297.105..60.39.30.24.20.17.16.14.13.12.11.10.10..9..9..9..9..9..9..9..9..9..9
.385.137..73.50.36.29.24.21.18.17.15.14.13.12.11.11.10.10.10.10.10.10.10.10.10
.490.165..88.57.42.32.27.23.20.18.17.15.14.13.12.11.11.10.10.10.10.10.10.10.10
.627.210.110.70.50.40.32.27.24.21.19.18.16.15.14.13.12.12.11.11.11.11.11.11.11
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LINKS
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FORMULA
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G.f. of column k: x^(2-2*k) * Product_{j>=k} 1/(1-x^j). - Alois P. Heinz, Nov 06 2011
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EXAMPLE
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All solutions for n=5, k=3: 3+3+3, 3+6, 4+5, 9.
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MAPLE
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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):
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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