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

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

1,6

Links

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

Formula

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

Example

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)

Maple

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

Crossrefs

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

Sequence in context: A181850 A335050 A062761 * A308513 A045683 A343031

Adjacent sequences: A117521 A117522 A117523 * A117525 A117526 A117527

Keyword

easy,nonn

Author

Vladeta Jovovic, Apr 26 2006

Status

approved