A116635 - OEIS

%I #21 Jan 23 2025 08:32:06

%S 0,0,0,1,1,2,5,7,11,19,27,40,61,85,120,170,232,316,433,576,767,1017,

%T 1332,1735,2259,2905,3730,4768,6058,7663,9676,12137,15191,18945,23541,

%U 29150,36026,44336,54453,66686,81456,99227,120653,146275,177015,213724

%N Number of parts that are multiples of 3 in all partitions of n.

%H G. C. Greubel, <a href="/A116635/b116635.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..floor(n/3)} k*A116633(n,k).

%F G.f.: (Sum_{i>=1} x^(3*i)/(1-x^(3*i)))/(Product_{j>=1} (1-x^j)).

%e a(6)=5 because in the 11 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 5 multiples of 3 (shown between parentheses).

%p g:=sum(x^(3*i)/(1-x^(3*i)),i=1..50)/product(1-x^j,j=1..50): gser:=series(g,x=0,60): seq(coeff(gser,x,n),n=0..52);

%t With[{nmax = 50}, CoefficientList[Series[Sum[x^(3*k)/(1 - x^(3*k)), {k, 1, nmax}]/Product[(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]] (* _G. C. Greubel_, Nov 19 2017 *)

%Y Cf. A116633.

%K nonn,changed

%O 0,6

%A _Emeric Deutsch_, Feb 19 2006