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%I #41 Oct 27 2023 19:12:24
%S 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,
%T 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,
%U 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
%N T(n,k) = Number of partitions of n+2k-2 into parts >= k.
%C Row n goes to floor(n/2)+1
%C Table starts
%C ...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
%C ...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
%C ...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
%C ...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
%C ...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
%C ..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
%C ..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
%C ..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
%C ..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
%C ..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
%C ..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
%C ..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
%C .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
%C .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
%C .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
%C .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
%C .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
%C .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
%C .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
%C .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
%H R. H. Hardin and Alois P. Heinz, <a href="/A199086/b199086.txt">Table of n, a(n) for n = 1..10011</a>
%F G.f. of column k: x^(2-2*k) * Product_{j>=k} 1/(1-x^j). - _Alois P. Heinz_, Nov 06 2011
%e All solutions for n=5, k=3: 3+3+3, 3+6, 4+5, 9.
%p b:= proc(n, i) option remember;
%p if n<0 then 0
%p elif n=0 then 1
%p elif i>n then 0
%p else b(n-i, i) +b(n, i+1)
%p fi
%p end:
%p T:= (n, k)-> b(n+2*k-2, k):
%p seq(seq(T(n, d+1-n), n=1..d), d=1..20); # _Alois P. Heinz_, Nov 06 2011
%t 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_ *)
%Y Column 1 is A000041
%Y Column 2 is A002865(n+2)
%Y Column 3 is A008483(n+4)
%Y Column 4 is A008484(n+6)
%Y Column 5 is A026798(n+13)
%Y Column 6 is A026799(n+16)
%Y Column 7 is A026800(n+19)
%Y Column 8 is A026801(n+22)
%Y Column 9 is A026802(n+25)
%Y Column 10 is A026803(n+28)
%K nonn,tabl
%O 1,3
%A _R. H. Hardin_, Nov 06 2011
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