A075196 - OEIS

%I #24 Mar 13 2016 17:38:27

%S 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,

%T 330,149,15,8,56,238,660,1072,906,298,22,9,72,364,1260,2777,3622,2367,

%U 591,30,10,90,528,2198,6174,11160,11676,6027,1132,42,11,110,735,3582,12292,28784,42805,36450,14873,2139,56

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

%C 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

%H Alois P. Heinz, <a href="/A075196/b075196.txt">Rows n = 1..141, flattened</a>

%F T(n,k) = Sum_{i=0..k} C(k,i) * A255903(n,i). - _Alois P. Heinz_, Mar 10 2015

%e 1,  2,   3,    4,    5, ...

%e 2,  6,  12,   20,   30, ...

%e 3, 14,  38,   80,  145, ...

%e 5, 33, 117,  305,  660, ...

%e 7, 70, 330, 1072, 2777, ...

%p with(numtheory):

%p A:= proc(n, k) option remember; local d, j;

%p       `if`(n=0, 1, add(add(d*binomial(d+k-1, k-1),

%p        d=divisors(j)) *A(n-j, k), j=1..n)/n)

%p     end:

%p seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # _Alois P. Heinz_, Sep 26 2012

%t 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 *)

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

%Y Rows 1-3: A000027, A002378, A162147.

%Y Main diagonal: A075197.

%Y Cf. A255903.

%K nonn,tabl

%O 1,2

%A _Christian G. Bower_, Sep 07 2002