A015741 Number of 6's in all the partitions of n into distinct parts.

0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 12, 14, 17, 21, 24, 29, 34, 40, 47, 55, 65, 75, 88, 102, 118, 137, 157, 181, 208, 238, 272, 311, 355, 404, 460, 522, 592, 671, 758, 856, 966, 1088, 1224, 1377, 1546, 1734, 1944

Offset

1,9

Comments

a(n+6) = A015753(n). - Alois P. Heinz, Aug 24 2011

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Formula

G.f.: x^6 * product(j>=1, 1+x^j )/(1+x^6). - Emeric Deutsch, Apr 17 2006

Corresponding g.f. for "number of k's" is x^k/(1+x^k)*prod(n>=1, 1+x^n ). [Joerg Arndt, Feb 20 2014]

a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015

Example

a(9) = 2 because in the 8 (=A000009(9)) partitions of 9 into distinct parts, namely [9], [8,1], [7,2], [6,3], [6,2,1], [5,4], [5,3,1] and [4,3,2] we have altogether two parts equal to 6.

Maple

g:=x^6*product(1+x^j, j=1..60)/(1+x^6): gser:=series(g, x=0, 57): seq(coeff(gser, x, n), n=1..54); # Emeric Deutsch, Apr 17 2006

Mathematica

nmax = 100; Rest[CoefficientList[Series[x^6/(1+x^6) * Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 30 2015 *)
Table[Count[Flatten@Select[IntegerPartitions[n], DeleteDuplicates[#] == # &], 6], {n, 54}] (* Robert Price, May 16 2020 *)

Crossrefs

Cf. A015753.

Sequence in context: A261770 A096792 A335746 * A015753 A005686 A328675

Adjacent sequences: A015738 A015739 A015740 * A015742 A015743 A015744

Keyword

nonn

Author

Clark Kimberling

Status

approved