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).

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

0,5

Comments

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

Links

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

Formula

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

Example

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).

Maple

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);

Mathematica

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 *)

Crossrefs

Cf. A116644. Column k=2 of A197126.

Sequence in context: A116651 A135586 A168542 * A306318 A091188 A147678

Adjacent sequences: A116643 A116644 A116645 * A116647 A116648 A116649

Keyword

nonn

Author

Emeric Deutsch, Feb 20 2006

Status

approved