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

1,3

Comments

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

Links

R. H. Hardin and Alois P. Heinz, Table of n, a(n) for n = 1..10011

Formula

G.f. of column k: x^(2-2*k) * Product_{j>=k} 1/(1-x^j). - Alois P. Heinz, Nov 06 2011

Example

All solutions for n=5, k=3: 3+3+3, 3+6, 4+5, 9.

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 *)

Crossrefs

Column 1 is A000041

Column 2 is A002865(n+2)

Column 3 is A008483(n+4)

Column 4 is A008484(n+6)

Column 5 is A026798(n+13)

Column 6 is A026799(n+16)

Column 7 is A026800(n+19)

Column 8 is A026801(n+22)

Column 9 is A026802(n+25)

Column 10 is A026803(n+28)

Sequence in context: A275723 A198338 A357554 * A098053 A272907 A128117

Adjacent sequences: A199083 A199084 A199085 * A199087 A199088 A199089

Keyword

nonn,tabl

Author

R. H. Hardin, Nov 06 2011

Status

approved