A116635 Number of parts that are multiples of 3 in all partitions of n.

0, 0, 0, 1, 1, 2, 5, 7, 11, 19, 27, 40, 61, 85, 120, 170, 232, 316, 433, 576, 767, 1017, 1332, 1735, 2259, 2905, 3730, 4768, 6058, 7663, 9676, 12137, 15191, 18945, 23541, 29150, 36026, 44336, 54453, 66686, 81456, 99227, 120653, 146275, 177015, 213724

Offset

0,6

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

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

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

Example

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).

Maple

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);

Mathematica

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 *)

Crossrefs

Cf. A116633.

Sequence in context: A251964 A045351 A051645 * A340759 A252280 A184774

Adjacent sequences: A116632 A116633 A116634 * A116636 A116637 A116638

Keyword

nonn

Author

Emeric Deutsch, Feb 19 2006

Status

approved