A015743 Number of 8's in all the partitions of n into distinct parts.

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 10, 13, 15, 18, 22, 27, 31, 37, 44, 51, 61, 71, 82, 95, 111, 128, 148, 171, 195, 225, 258, 295, 337, 384, 437, 497, 565, 639, 724, 818, 923, 1042, 1173, 1319, 1483, 1665

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Examples

			a(11)=2 because in the 12 (=A000009(11)) partitions of 11 into distinct parts, namely [11], [10,1], [9,2], [8,3], [8,2,1], [7,4], [7,3,1], [6,5], [6,4,1], [6,3,2], [5,4,2] and [5,3,2,1], altogether we have two parts equal to 8.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 2000 terms from Robert Price)

Programs

Maple

g:=x^8*product(1+x^j,j=1..60)/(1+x^8): gser:=series(g,x=0,57): seq(coeff(gser,x,n),n=1..54); # Emeric Deutsch, Apr 17 2006

Mathematica

Table[Count[Flatten@Select[IntegerPartitions[n], DeleteDuplicates[#] == # &], 8], {n, 54}] (* Robert Price, Jun 13 2020 *)
nmax = 100; Rest[CoefficientList[Series[x^8/(1+x^8) * Product[(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 15 2025 *)

Formula

G.f.: x^8*Product_{j>=1} (1+x^j)/(1+x^8). - Emeric Deutsch, Apr 17 2006

a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jun 15 2025