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

Author

Clark Kimberling

Keywords

• nonn

Examples

			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.
		

Links

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

Programs

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 *)

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