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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.
9
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, 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, 42, 1, 1, 2, 3, 5, 7, 10, 12, 15, 15, 12, 56
OFFSET
0,6
LINKS
FORMULA
G.f. of column 0: Product_{m>0} 1/(1-x^m).
G.f. of column k>0: (1-x^k) * Product_{m>0} 1/(1-x^m).
A(n,0) = A000041(n); A(n,k) = A000041(n) - A000041(n-k) for k>0.
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
EXAMPLE
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 0, 1, 1, 1, 1, ...
2, 1, 1, 2, 2, 2, ...
3, 1, 2, 2, 3, 3, ...
5, 2, 3, 4, 4, 5, ...
7, 2, 4, 5, 6, 6, ...
MAPLE
A41:= n-> `if`(n<0, 0, combinat[numbpart](n)):
A:= (n, k)-> A41(n) -`if`(k>0, A41(n-k), 0):
seq(seq(A(n, d-n), n=0..d), d=0..11);
MATHEMATICA
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 *)
CROSSREFS
Rows n=0-1 give: A000012, A060576.
Main diagonal gives A000065 (for n>0).
Sequence in context: A124944 A094392 A111946 * A237513 A137844 A263845
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Dec 04 2010
STATUS
approved