A116644 -id:A116644 - cp's OEIS frontend

A116645 Number of partitions of n having no doubletons. By a doubleton in a partition we mean an occurrence of a part exactly twice (the partition [4,(3,3),2,2,2,(1,1)] of 18 has two doubletons, shown between parentheses).

1, 1, 1, 3, 3, 5, 8, 10, 13, 20, 26, 33, 46, 58, 75, 101, 125, 157, 206, 253, 317, 403, 494, 608, 760, 926, 1131, 1393, 1685, 2038, 2487, 2985, 3585, 4331, 5168, 6172, 7392, 8771, 10410, 12382, 14622, 17258, 20400, 23975, 28159, 33115, 38739, 45298, 53000

Offset: 0

Emeric Deutsch and Vladeta Jovovic, Feb 20 2006

nonn

Number of partitions of n having no part that appears exactly twice.

Infinite convolution product of [1,1,0,1,1,1,1,1,1,1] aerated n-1 times. I.e., [1,1,0,1,1,1,1,1,1,1] * [1,0,1,0,0,0,1,0,1,0] * [1,0,0,1,0,0,0,0,0,1] * ... - Mats Granvik, Gary W. Adamson, Aug 07 2009

			a(4) = 3 because we have [4],[3,1] and [1,1,1,1] (the partitions [2,2] and [2,1,1] do not qualify since each of them has a doubleton).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column 0 of A116644.

Cf. A000041, A007690, A116595, A183559, A183568.

h:=product((1-x^(2*j)+x^(3*j))/(1-x^j),j=1..60): hser:=series(h,x=0,60): seq(coeff(hser,x,n),n=0..56);

nn=48;CoefficientList[Series[Product[1/(1-x^i)-x^(2i),{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 30 2013 *)

G.f.: Product_{j>=1} (1-x^(2j)+x^(3j))/(1-x^j).
G.f. for the number of partitions of n having no part that appears exactly m times is Product_{k>0} (1/(1-x^k)-x^(m*k)).
a(n) = A000041(n) - A183559(n) = A183568(n,0) - A183568(n,2). - Alois P. Heinz, Oct 09 2011
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-2*x) + exp(-4*x)) dx = 0.64673501839556449802623523266221107725058748270577037891948... - Vaclav Kotesovec, Jun 12 2025

A116646 Number of doubletons in all partitions of n. By a doubleton in a partition we mean an occurrence of a part exactly twice (the partition [4,(3,3),2,2,2,(1,1)] has two doubletons, shown between parentheses).

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 4, 5, 10, 11, 20, 25, 38, 49, 73, 91, 131, 167, 228, 291, 392, 493, 653, 822, 1065, 1336, 1714, 2131, 2706, 3354, 4209, 5193, 6471, 7934, 9817, 11990, 14725, 17909, 21875, 26477, 32172, 38797, 46893, 56339, 67804, 81147, 97260, 116017

Offset: 0

Emeric Deutsch, Feb 20 2006

nonn

a(n) = (the number of 2's in all partitions of n) - (the number of 3's in all partitions of n). - Gregory L. Simay, Jul 28 2020

			a(6) = 4 because in the partitions of 6, namely [6],[5,1],[4,2],[4,(1,1)],[(3,3)],[3,2,1],[3,1,1,1],[2,2,2],[(2,2),(1,1)],[2,1,1,1,1] and [1,1,1,1,1,1], we have a total of 4 doubletons (shown between parentheses).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A116644. Column k=2 of A197126.

f:= x^2/(1+x)/(1-x^3)/product(1-x^j, j=1..70): fser:= series(f,x=0,70): seq(coeff(fser,x,n), n=0..55);

nmax = 50; CoefficientList[Series[x^2/((1+x)*(1-x^3)) * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)
Table[Sum[PartitionsP[n-6*m-2] - PartitionsP[n-6*m-3] + PartitionsP[n-6*m-4], {m, 0, Floor[n/6]}], {n, 0, 50}] (* Vaclav Kotesovec, Mar 07 2016 *)

G.f.: x^2 / ((1+x)*(1-x^3)*(Product_{j>=1} 1-x^j)).
a(n) = Sum_{k>=0} k * A116644(n,k).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (3*2^(5/2)*Pi*sqrt(n)). - Vaclav Kotesovec, Mar 07 2016

A118806 Triangle read by rows: T(n,k) is the number of partitions of n having k parts of multiplicity 3 (n,k>=0).

Original entry on oeis.org

1, 1, 2, 2, 1, 5, 6, 1, 9, 2, 12, 3, 19, 3, 24, 5, 1, 34, 8, 43, 13, 62, 13, 2, 77, 23, 1, 105, 28, 2, 132, 40, 4, 177, 49, 5, 220, 71, 6, 287, 89, 8, 1, 356, 123, 11, 462, 147, 18, 570, 198, 23, 1, 723, 249, 29, 1, 888, 329, 37, 1, 1121, 400, 50, 4, 1370, 518, 69, 1, 1705, 642, 85

Offset: 0

Emeric Deutsch, Apr 29 2006

T(n,0)=A118807(n). T(n,1)=A118808(n). Row sums yield the partition numbers (A000041). Sum(k*T(n,k), k>=0)=A117524(n) (n>=1).

			T(12,2) = 2 because we have [3,3,3,1,1,1] and [3,2,2,2,1,1,1].
Triangle starts:
1;
1;
2;
2,  1;
5;
6,  1;
9,  2;
12, 3;

Alois P. Heinz, Rows n = 0..703

Cf. A000041, A118807, A118808, A117524, A116595, A116644.

g:=product(1+x^j+x^(2*j)+t*x^(3*j)+x^(4*j)/(1-x^j),j=1..35): gser:=simplify(series(g,x=0,35)): P[0]:=1: for n from 1 to 30 do P[n]:=coeff(gser,x^n) od: for n from 0 to 30 do seq(coeff(P[n],t,j),j=0..degree(P[n])) od; # sequence given in triangular form

G.f.: product(1+x^j+x^(2j)+tx^(3j)+x^(4j)/(1-x^j), j=1..infinity).

cp's OEIS Frontend

A116645 Number of partitions of n having no doubletons. By a doubleton in a partition we mean an occurrence of a part exactly twice (the partition [4,(3,3),2,2,2,(1,1)] of 18 has two doubletons, shown between parentheses).

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A116646 Number of doubletons in all partitions of n. By a doubleton in a partition we mean an occurrence of a part exactly twice (the partition [4,(3,3),2,2,2,(1,1)] has two doubletons, shown between parentheses).

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A118806 Triangle read by rows: T(n,k) is the number of partitions of n having k parts of multiplicity 3 (n,k>=0).

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