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%I #14 May 24 2018 04:22:11
%S 0,0,1,0,1,2,3,3,7,8,13,17,25,32,48,59,83,108,145,183,247,310,406,512,
%T 659,824,1055,1307,1651,2047,2558,3146,3913,4788,5904,7202,8821,10707,
%U 13054,15770,19118,23027,27775,33312,40029,47835,57231,68182,81261
%N Total number of parts of multiplicity 3 in all partitions of n.
%H Alois P. Heinz, <a href="/A117524/b117524.txt">Table of n, a(n) for n = 1..1000</a>
%F 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).
%F a(n) = Sum(k*A118806(n,k), k>=0). - _Emeric Deutsch_, Apr 29 2006
%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (24*Pi*sqrt(2*n)). - _Vaclav Kotesovec_, May 24 2018
%e 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)
%p 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
%Y Cf. A024786, A116646. Column k=3 of A197126.
%K easy,nonn
%O 1,6
%A _Vladeta Jovovic_, Apr 26 2006
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