A118806 -id:A118806 - cp's OEIS frontend

A118807 Number of partitions of n having no parts with multiplicity 3.

1, 1, 2, 2, 5, 6, 9, 12, 19, 24, 34, 43, 62, 77, 105, 132, 177, 220, 287, 356, 462, 570, 723, 888, 1121, 1370, 1705, 2074, 2570, 3111, 3816, 4601, 5617, 6743, 8170, 9777, 11794, 14058, 16858, 20029, 23932, 28334, 33692, 39772, 47133, 55468, 65471, 76840

Offset: 0

Emeric Deutsch, Apr 29 2006

nonn

Column 0 of A118806.

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

			a(6) = 9 because among the 11 (=A000041(6)) partitions of 6 only [2,2,2] and [3,1,1,1] have parts with multiplicity 3.

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

Cf. A000041, A118806, A007690, A116645, A183560, A183568.

g:=product(1+x^j+x^(2*j)+x^(4*j)/(1-x^j),j=1..60): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..50);

nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k) + x^(4*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)

G.f.: Product_{j>=1} (1 + x^j + x^(2j) + x^(4j)/(1-x^j)).
a(n) = A000041(n) - A183560(n) = A183568(n,0) - A183568(n,3). - 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(-3*x) + exp(-5*x)) dx = 0.73597677748514060768682570953508781551028221145343244320009... - Vaclav Kotesovec, Jun 12 2025

A117524 Total number of parts of multiplicity 3 in all partitions of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 3, 7, 8, 13, 17, 25, 32, 48, 59, 83, 108, 145, 183, 247, 310, 406, 512, 659, 824, 1055, 1307, 1651, 2047, 2558, 3146, 3913, 4788, 5904, 7202, 8821, 10707, 13054, 15770, 19118, 23027, 27775, 33312, 40029, 47835, 57231, 68182, 81261

Offset: 1

Vladeta Jovovic, Apr 26 2006

			a(9) = 7 because among the 30 (=A000041(9)) partitions of 9 only [6,(1,1,1)],[4,2,(1,1,1)],[(3,3,3)],[3,3,(1,1,1)],[3,(2,2,2)],[(2,2,2),(1,1,1)] contain parts of multiplicity 3 and their total number is 7 (shown between parentheses)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A024786, A116646. Column k=3 of A197126.

g:=(x^3/(1-x^3)-x^4/(1-x^4))/product(1-x^i,i=1..65): gser:=series(g,x=0,62): seq(coeff(gser,x,n),n=1..58); # Emeric Deutsch, Apr 29 2006

G.f. for total number of parts of multiplicity m in all partitions of n is (x^m/(1-x^m)-x^(m+1)/(1-x^(m+1)))/Product(1-x^i,i=1..infinity).
a(n) = Sum(k*A118806(n,k), k>=0). - Emeric Deutsch, Apr 29 2006
a(n) ~ exp(Pi*sqrt(2*n/3)) / (24*Pi*sqrt(2*n)). - Vaclav Kotesovec, May 24 2018

A118808 Number of partitions of n having exactly one part with multiplicity 3.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 3, 3, 5, 8, 13, 13, 23, 28, 40, 49, 71, 89, 123, 147, 198, 249, 329, 400, 518, 642, 825, 996, 1265, 1545, 1941, 2340, 2920, 3533, 4357, 5233, 6417, 7717, 9399, 11211, 13591, 16215, 19540, 23189, 27826, 32990, 39392, 46504, 55313, 65200

Offset: 0

Emeric Deutsch, Apr 29 2006

nonn

Column 1 of A118806.

			a(9)=5 because we have [6,1,1,1],[4,2,1,1,1],[3,3,3],[3,3,1,1,1] and [3,2,2,2].

Cf. A118806, A118807, A116596.

g:=product((1-x^(3*j)+x^(4*j))/(1-x^j),j=1..70)*sum(x^(3*j)*(1-x^j)/(1-x^(3*j)+x^(4*j)),j=1..70): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..60);

Table[Length[Select[IntegerPartitions[n],Count[Length/@Split[#], 3]==1&]],{n,0,60}]  (* Harvey P. Dale, Mar 24 2011 *)

G.f.=product([1-x^(3j)+x^(4j)]/(1-x^j), j=1..infinity)*sum(x^(3j)*(1-x^j)/[1-x^(3j)+x^(4j)], j=1..infinity).

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A118807 Number of partitions of n having no parts with multiplicity 3.

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A117524 Total number of parts of multiplicity 3 in all partitions of n.

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A118808 Number of partitions of n having exactly one part with multiplicity 3.

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