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A175788 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of n that do not contain k as a part.

%I A175788 #26 Oct 04 2018 20:04:38
%S A175788 1,1,1,1,0,2,1,1,1,3,1,1,1,1,5,1,1,2,2,2,7,1,1,2,2,3,2,11,1,1,2,3,4,4,
%T A175788 4,15,1,1,2,3,4,5,6,4,22,1,1,2,3,5,6,8,8,7,30,1,1,2,3,5,6,9,10,11,8,
%U A175788 42,1,1,2,3,5,7,10,12,15,15,12,56
%N A175788 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of n that do not contain k as a part.
%H A175788 Alois P. Heinz, <a href="/A175788/b175788.txt">Antidiagonals n = 0..140, flattened</a>
%F A175788 G.f. of column 0: Product_{m>0} 1/(1-x^m).
%F A175788 G.f. of column k>0: (1-x^k) * Product_{m>0} 1/(1-x^m).
%F A175788 A(n,0) = A000041(n); A(n,k) = A000041(n) - A000041(n-k) for k>0.
%F A175788 For fixed k>0, A(n,k) ~ k*Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + k*Pi/(2*sqrt(6)))/sqrt(n) + (1/8 + 3*k/2 + 9/(2*Pi^2) + Pi^2/6912 + k*Pi^2/288 + k^2*Pi^2/36)/n). - _Vaclav Kotesovec_, Nov 04 2016
%e A175788 Square array A(n,k) begins:
%e A175788   1, 1, 1, 1, 1, 1, ...
%e A175788   1, 0, 1, 1, 1, 1, ...
%e A175788   2, 1, 1, 2, 2, 2, ...
%e A175788   3, 1, 2, 2, 3, 3, ...
%e A175788   5, 2, 3, 4, 4, 5, ...
%e A175788   7, 2, 4, 5, 6, 6, ...
%p A175788 A41:= n-> `if`(n<0, 0, combinat[numbpart](n)):
%p A175788 A:= (n,k)-> A41(n) -`if`(k>0, A41(n-k), 0):
%p A175788 seq(seq(A(n,d-n), n=0..d), d=0..11);
%t A175788 A41[n_] := If[n<0, 0, PartitionsP[n]]; A[n_, k_] := A41[n]-If[k>0, A41[n-k], 0]; Table[A[n, d-n], {d, 0, 11}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 18 2017, translated from Maple *)
%Y A175788 Columns k=0-10 give: A000041, A002865, A027336, A027337, A027338, A027339, A027340, A027341, A027342, A027343, A027344.
%Y A175788 Rows n=0-1 give: A000012, A060576.
%Y A175788 Main diagonal gives A000065 (for n>0).
%K A175788 nonn,tabl
%O A175788 0,6
%A A175788 _Alois P. Heinz_, Dec 04 2010