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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 20 2006
STATUS
approved