A027343 -id:A027343 - 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

A343669 Number of partitions of an n-set without blocks of size 9.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21146, 115965, 678460, 4212497, 27633712, 190795218, 1381942530, 10470109267, 82764226404, 681048663329, 5822029128397, 51610194855972, 473631475252041, 4492967510009533, 43996047374513046, 444151309687221889, 4617189912288741028

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000110, A000296, A027343, A097514, A124504, A343664, A343665, A343666, A343667, A343668.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=9, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^9/9!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 9 k]/((n - 9 k)! k! (9!)^k), {k, 0, Floor[n/9]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 9, 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^9/9!).

a(n) = n! * Sum_{k=0..floor(n/9)} (-1)^k * Bell(n-9*k) / ((n-9*k)! * k! * (9!)^k).

A027344 Number of partitions of n that do not contain 10 as a part.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 75, 98, 130, 169, 220, 282, 363, 460, 585, 736, 925, 1154, 1440, 1782, 2205, 2713, 3333, 4075, 4977, 6050, 7347, 8888, 10735, 12925, 15541, 18627, 22297, 26620, 31734, 37741, 44825, 53118, 62865

Offset: 0

Clark Kimberling

nonn

10th column of A175788. Cf. A000041, A027336, A027337-A027343.

A41:= n-> `if`(n<0, 0, combinat[numbpart](n)):
a:= n-> A41(n) -A41(n-10):
seq(a(n), n=0..50);

G.f.: (1-x^10) Product_{m>0} 1/(1-x^m).
a(n) = A000041(n)-A000041(n-10).
a(n) ~ 5*Pi * exp(sqrt(2*n/3)*Pi) / (6*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 10*Pi/(2*sqrt(6)))/sqrt(n) + (121/8 + 9/(2*Pi^2) + 19441*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

More terms from Benoit Cloitre, Dec 10 2002

Edited by Alois P. Heinz, Dec 04 2010

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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A343669 Number of partitions of an n-set without blocks of size 9.

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A027344 Number of partitions of n that do not contain 10 as a part.

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