A027337 -id:A027337 - cp's OEIS frontend

A027336 Number of partitions of n that do not contain 2 as a part.

1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 35, 45, 58, 75, 96, 121, 154, 193, 242, 302, 375, 463, 573, 703, 861, 1052, 1282, 1555, 1886, 2277, 2745, 3301, 3961, 4740, 5667, 6754, 8038, 9548, 11323, 13398, 15836, 18678, 22001, 25873, 30383, 35620, 41715, 48771

Offset: 0

Clark Kimberling

nonn

Pairwise sums of sequence A002865 (partitions in which the least part is at least 2).
Also number of partitions of n into parts with at most one 1. - Reinhard Zumkeller, Oct 25 2004
Also number of partitions of n into parts with at least half of the parts having size 1; equivalently (by duality) number of partitions of n where the large part is at least twice as big as the second largest part. - Franklin T. Adams-Watters, Jun 08 2005
Also number of 2-regular not necessarily connected graphs with loops allowed but no multiple edges. - Jason Kimberley, Jan 05 2011

Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
V. Jelinek, T. Mansour, and M. Shattuck, On multiple pattern avoiding set partitions, Advances in Applied Mathematics Volume 50, Issue 2, February 2013, Pages 292-326. - _N. J. A. Sloane_, Jan 01 2013
Jerome Kelleher and Barry O'Sullivan, Generating All Partitions: A Comparison Of Two Encodings, arXiv:0909.2331 [cs.DS], 2009-2014. [_Peter Luschny_, Oct 24 2010]
Krishna Menon and Anurag Singh, Pattern avoidance and dominating compositions, arXiv:2104.07274 [math.CO], 2021.
Mircea Merca, Fast algorithm for generating ascending compositions, arXiv:1903.10797 [math.CO], 2019.

Cf. A000041, A002865, A027337.
2-regular not necessarily connected graphs: A008483 (simple graphs), A000041 (multigraphs with loops allowed), A002865 (multigraphs with loops forbidden), A027336 (graphs with loops allowed but no multiple edges). - Jason Kimberley, Jan 05 2011
Column k=1 of A292622.

A41 := func;
[A41(n)-A41(n-2):n in [0..49]]; // Jason Kimberley, Jan 05 2011

with(combinat): a:=proc(n) if n=0 then 1 elif n=1 then 1 else numbpart(n)-numbpart(n-2) fi end: seq(a(n),n=0..49); # Emeric Deutsch, Feb 18 2006

a[n_] = PartitionsP[n] - PartitionsP[n-2]; a /@ Range[0, 49] (* Jean-François Alcover, Jul 13 2011, after Emeric Deutsch *)

a(n)=if(n<0,0,polcoeff((1-x^2)/eta(x+x*O(x^n)),n))

G.f.: (1 - x^2)*Product_{m>=1} 1/(1 - x^m).
a(n) = A000041(n) - A000041(n-2).
a(n) = p(n) - p(n-2) for n >= 2, where p(n) are the partition numbers (A000041); follows at once from the g.f. - Emeric Deutsch, Feb 18 2006
a(n) ~ exp(sqrt(2*n/3)*Pi)*Pi / (6*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + 25*Pi/(24*sqrt(6)))/sqrt(n) + (25/8 + 9/(2*Pi^2) + 817*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

More terms from Benoit Cloitre, Dec 10 2002

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.

Original entry on oeis.org

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

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

A263232 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 3 (n >= 0, 0 <= k <= floor(n/3)).

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 1, 5, 2, 8, 2, 1, 10, 4, 1, 15, 5, 2, 19, 8, 2, 1, 27, 10, 4, 1, 34, 15, 5, 2, 47, 19, 8, 2, 1, 59, 27, 10, 4, 1, 79, 34, 15, 5, 2, 99, 47, 19, 8, 2, 1, 130, 59, 27, 10, 4, 1, 162, 79, 34, 15, 5, 2, 209, 99, 47, 19, 8, 2, 1, 259, 130, 59, 27, 10, 4, 1

Offset: 0

Emeric Deutsch, Nov 01 2015

Row n has 1+floor(n/3) terms. Row sums are the partition numbers (A000041). T(n,0) = A027337(n). Sum_{k=0..floor(n/3)} k*T(n,k) = A024787(n).

			T(7,1) = 4 because we have [4,3], [3,2,2], [3,2,1,1], and [3,1,1,1,1].
T(9,2) = 2 because we have [3,3,2,1] and [3,3,1,1,1].
Triangle starts:
  1;
  1;
  2;
  2, 1;
  4, 1;
  5, 2;
  8, 2, 1.

Alois P. Heinz, Rows n = 0..300, flattened

Cf. A000041, A027337, A024787, A116599.

g := 1/((1-x)*(1-x^2)*(1-t*x^3)*(product(1-x^j, j = 4 .. 80))): gser := simplify(series(g, x = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, `if`(i=3, x, 1)*
      `if`(i>n, 0, b(n-i, i)) +b(n, i-1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..25);  # Alois P. Heinz, Nov 01 2015

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, If[i == 3, x, 1]* If[i > n, 0, b[n - i, i]] + b[n, i - 1]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, Jan 21 2016, after Alois P. Heinz *)

G.f.: (1-x)*(1-x^2)*(1-tx^3)*Product_{j>=4} (1-x^j).

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A027336 Number of partitions of n that do not contain 2 as a part.

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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.

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

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

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Maple

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Extensions

A263232 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 3 (n >= 0, 0 <= k <= floor(n/3)).

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