A274109 -id:A274109 - cp's OEIS frontend

A216665 Triangular array read by rows: T(n,k) is the number of partitions of n into k parts of 2 different sizes; n>=3, 2<=k<=n-1.

1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 3, 3, 2, 2, 1, 3, 3, 2, 2, 2, 1, 4, 3, 2, 3, 2, 2, 1, 4, 4, 5, 1, 3, 2, 2, 1, 5, 5, 3, 4, 2, 3, 2, 2, 1, 5, 4, 4, 3, 3, 2, 3, 2, 2, 1, 6, 6, 4, 5, 2, 4, 2, 3, 2, 2, 1, 6, 6, 7, 5, 5, 1, 4, 2, 3, 2, 2, 1, 7, 6, 4, 3, 4, 4, 2, 4, 2, 3, 2, 2, 1

Offset: 3

Geoffrey Critzer, Sep 13 2012

Row sums = A002133.

First column (corresponding to k=2) = floor( (n-1)/2 ).

			T(8,3) = 3 because we have: 6+1+1, 4+2+2, 3+3+2.
Triangle indexed from n=3 and k=2:
1;
1, 1;
2, 2, 1;
2, 1, 2, 1;
3, 3, 2, 2, 1;
3, 3, 2, 2, 2, 1;
4, 3, 2, 3, 2, 2, 1;
4, 4, 5, 1, 3, 2, 2, 1;

Alois P. Heinz, Rows n = 3..143, flattened
N. B. Tani and S. Bouroubi, Enumeration of the Partitions of an Integer into Parts of a Specified Number of Different Sizes and Especially Two Sizes, J. Integer Seqs., Vol. 14 (2011), #11.3.6. (This sequence is Table 1 on p. 10).

Cf. A117955, A117956, A274108, A274109.

nn=15;ss=Sum[Sum[y^2 x^(i+j)/(1-y x^i)/(1-y x^j),{j,1,i-1}],{i,1,nn}]; f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[ss,{x,0,nn}], {x,y}]]//Flatten

G.f.: Sum_{i>=1} Sum_{j=1..n-1} y^2*x^(i+j)/((1-y*x^j)*(1-y*x^i)).

A274108 Number of partitions of n into parts with exactly two different sizes, the sizes being relatively prime.

Original entry on oeis.org

0, 0, 1, 2, 5, 5, 11, 11, 16, 17, 27, 21, 37, 33, 38, 42, 59, 46, 71, 57, 70, 75, 97, 72, 104, 99, 109, 103, 141, 102, 157, 133, 148, 153, 166, 140, 207, 183, 192, 174, 241, 180, 259, 215, 223, 247, 295, 219, 300, 260, 292, 279, 353, 275, 336, 300, 346, 351

Offset: 1

N. J. A. Sloane, Jun 17 2016

			Explanation of a(3)-a(6):
n=3: 21
n=4: 31, 211
n=5: 41, 32, 311, 221, 2111
n=6: 51, 411, 3111, 2211, 21111

Alois P. Heinz, Table of n, a(n) for n = 1..10000
N. Benyahia Tani, S. Bouroubi, and O. Kihel, An effective approach for integer partitions using exactly two distinct sizes of parts, Bulletin du Laboratoire 03 (2015), 18-27.

Row sums of triangle in A274109.

Cf. A002133, A117955, A117956, A216665.

seq(n)={my(v=Vec(sum(k=1, n-1, numdiv(k)*x^k, O(x^n))^2, -n), u=vector(n, n, moebius(n))); dirmul(u, vector(#v, n, v[n]+numdiv(n)-sigma(n))/2)} \\ Andrew Howroyd, Nov 10 2024

from math import gcd
from sympy import divisors
def A274108(n): return sum(1 for ax in range(1,n-1) for a in divisors(ax,generator=True) for b in divisors(n-ax,generator=True) if aChai Wah Wu, Dec 11 2024

Formula

Moebius transform of A002133. - Andrew Howroyd, Nov 10 2024

Extensions

More terms from Alois P. Heinz, Jun 23 2016

cp's OEIS Frontend

A216665 Triangular array read by rows: T(n,k) is the number of partitions of n into k parts of 2 different sizes; n>=3, 2<=k<=n-1.

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