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

%I A075196 #26 Feb 20 2025 12:02:10
%S A075196 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 A075196 330,149,15,8,56,238,660,1072,906,298,22,9,72,364,1260,2777,3622,2367,
%U A075196 591,30,10,90,528,2198,6174,11160,11676,6027,1132,42,11,110,735,3582,12292,28784,42805,36450,14873,2139,56
%N A075196 Table T(n,k) by antidiagonals: T(n,k) = number of partitions of n balls of k colors.
%C A075196 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 A075196 Alois P. Heinz, <a href="/A075196/b075196.txt">Rows n = 1..141, flattened</a>
%F A075196 T(n,k) = Sum_{i=0..k} C(k,i) * A255903(n,i). - _Alois P. Heinz_, Mar 10 2015
%e A075196 Square array T(n,k) begins:
%e A075196   1,  2,   3,    4,    5, ...
%e A075196   2,  6,  12,   20,   30, ...
%e A075196   3, 14,  38,   80,  145, ...
%e A075196   5, 33, 117,  305,  660, ...
%e A075196   7, 70, 330, 1072, 2777, ...
%p A075196 with(numtheory):
%p A075196 A:= proc(n, k) option remember; local d, j;
%p A075196       `if`(n=0, 1, add(add(d*binomial(d+k-1, k-1),
%p A075196        d=divisors(j)) *A(n-j, k), j=1..n)/n)
%p A075196     end:
%p A075196 seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # _Alois P. Heinz_, Sep 26 2012
%t A075196 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 A075196 Columns 1-10: A000041, A005380, A217093, A255050, A255052, A270239, A270240, A270241, A270242, A270243.
%Y A075196 Rows 1-3: A000027, A002378, A162147.
%Y A075196 Main diagonal: A075197.
%Y A075196 Cf. A255903.
%K A075196 nonn,tabl
%O A075196 1,2
%A A075196 _Christian G. Bower_, Sep 07 2002