A027344 -id:A027344 - cp's OEIS frontend

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.

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

Alois P. Heinz, Dec 04 2010

			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, ...

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

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).

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);

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 *)

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

A027343 Number of partitions of n that do not contain 9 as a part.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 54, 74, 96, 128, 165, 216, 275, 355, 448, 571, 715, 901, 1120, 1399, 1727, 2139, 2625, 3228, 3938, 4812, 5840, 7094, 8568, 10352, 12447, 14967, 17919, 21450, 25581, 30496, 36234, 43031, 50951, 60292

Offset: 0

Clark Kimberling

nonn

9th column of A175788. Cf. A000041, A027336, A027337-A027344.

A41:= n-> `if`(n<0, 0, combinat[numbpart](n)):
a:= n-> A41(n) -A41(n-9):
seq(a(n), n=0..50);

Table[Count[IntegerPartitions[n],?(FreeQ[#,9]&)],{n,0,50}] (* _Harvey P. Dale, Oct 02 2022 *)

G.f.: (1-x^9) Product_{m>0} 1/(1-x^m).
a(n) = A000041(n)-A000041(n-9).
a(n) ~ 3*Pi * exp(sqrt(2*n/3)*Pi) / (4*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 9*Pi/(2*sqrt(6)))/sqrt(n) + (109/8 + 9/(2*Pi^2) + 15769*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

Edited by Alois P. Heinz, Dec 04 2010

A343671 Number of partitions of an n-set without blocks of size 10.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115974, 678559, 4213465, 27643007, 190884307, 1382802389, 10478516523, 82847813908, 681895648039, 5830788687491, 51702731250650, 474630475600569, 4503991075480297, 44120379612630694, 445584481578266277, 4634070027874688433

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000110, A000296, A027344, A097514, A124504, A343664, A343665, A343666, A343667, A343668, A343669.

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
       j=10, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 25 2023

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^10/10!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 10 k]/((n - 10 k)! k! (10!)^k), {k, 0, Floor[n/10]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 10, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

E.g.f.: exp(exp(x) - 1 - x^10/10!).

a(n) = n! * Sum_{k=0..floor(n/10)} (-1)^k * Bell(n-10*k) / ((n-10*k)! * k! * (10!)^k).

cp's OEIS Frontend

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.

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A027343 Number of partitions of n that do not contain 9 as a part.

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A343671 Number of partitions of an n-set without blocks of size 10.

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