A015740 Number of 5's in all the partitions of n into distinct parts.

0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 4, 6, 8, 9, 11, 14, 16, 19, 23, 27, 32, 38, 45, 53, 62, 72, 84, 97, 112, 130, 150, 172, 199, 228, 260, 298, 340, 386, 440, 500, 566, 642, 727, 820, 926, 1044, 1174, 1321, 1484, 1664, 1866, 2090

Offset

1,8

Links

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

Formula

G.f.: x^5*product(j>=1, 1+x^j )/(1+x^5). - 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 5.

Maple

g:=x^5*product(1+x^j, j=1..60)/(1+x^5): 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^5/(1+x^5) * Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 30 2015 *)
Table[Count[Flatten@Select[IntegerPartitions[n], DeleteDuplicates[#] == # &], 5], {n, 54}] (* Robert Price, May 16 2020 *)

Crossrefs

Sequence in context: A274759 A274157 A005863 * A015750 A275442 A303846

Adjacent sequences: A015737 A015738 A015739 * A015741 A015742 A015743

Keyword

nonn

Author

Clark Kimberling

Status

approved