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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

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-Fran├žois Alcover, Feb 18 2017, translated from Maple *)

CROSSREFS

Columns k=0-10 give: A000041, A002865, A027336, A027337, A027338, A027339, A027340, A027341, A027342, A027343, A027344.

Rows n=0-1 give: A000012, A060576.

Main diagonal gives A000065 (for n>0).

Sequence in context: A124944 A094392 A111946 * A237513 A137844 A263845

Adjacent sequences:  A175785 A175786 A175787 * A175789 A175790 A175791

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Dec 04 2010

STATUS

approved

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Last modified June 4 11:32 EDT 2020. Contains 334825 sequences. (Running on oeis4.)