A075196 Table T(n,k) by antidiagonals: T(n,k) = number of partitions of n balls of k colors.

1, 2, 2, 3, 6, 3, 4, 12, 14, 5, 5, 20, 38, 33, 7, 6, 30, 80, 117, 70, 11, 7, 42, 145, 305, 330, 149, 15, 8, 56, 238, 660, 1072, 906, 298, 22, 9, 72, 364, 1260, 2777, 3622, 2367, 591, 30, 10, 90, 528, 2198, 6174, 11160, 11676, 6027, 1132, 42, 11, 110, 735, 3582, 12292, 28784, 42805, 36450, 14873, 2139, 56

Offset

1,2

Comments

For k>=1, n->infinity is log(T(n,k)) ~ (1+1/k) * k^(1/(k+1)) * Zeta(k+1)^(1/(k+1)) * n^(k/(k+1)). - Vaclav Kotesovec, Mar 08 2015

Links

Alois P. Heinz, Rows n = 1..141, flattened

Formula

T(n,k) = Sum_{i=0..k} C(k,i) * A255903(n,i). - Alois P. Heinz, Mar 10 2015

Example

1,  2,   3,    4,    5, ...
2,  6,  12,   20,   30, ...
3, 14,  38,   80,  145, ...
5, 33, 117,  305,  660, ...
7, 70, 330, 1072, 2777, ...

Maple

with(numtheory):
A:= proc(n, k) option remember; local d, j;
      `if`(n=0, 1, add(add(d*binomial(d+k-1, k-1),
       d=divisors(j)) *A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 26 2012

Mathematica

Transpose[Table[nn=6; p=Product[1/(1- x^i)^Binomial[i+n, n], {i, 1, nn}]; Drop[CoefficientList[Series[p, {x, 0, nn}], x], 1], {n, 0, nn}]]//Grid  (* Geoffrey Critzer, Sep 27 2012 *)

Crossrefs

Columns 1-10: A000041, A005380, A217093, A255050, A255052, A270239, A270240, A270241, A270242, A270243.

Rows 1-3: A000027, A002378, A162147.

Main diagonal: A075197.

Cf. A255903.

Sequence in context: A183474 A294034 A210220 * A196912 A197079 A208340

Adjacent sequences: A075193 A075194 A075195 * A075197 A075198 A075199

Keyword

nonn,tabl

Author

Christian G. Bower, Sep 07 2002

Status

approved