A015737 Number of 3's in partitions of n into distinct parts.

0, 0, 1, 1, 1, 1, 1, 2, 3, 4, 4, 5, 6, 8, 10, 12, 14, 17, 20, 24, 29, 34, 40, 47, 55, 64, 75, 87, 101, 117, 135, 155, 179, 205, 235, 269, 307, 350, 399, 453, 514, 583, 660, 746, 843, 950, 1070, 1205, 1354, 1520, 1705, 1910, 2138, 2392

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Examples

			a(9) = 3 because in the eight 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], only three contain 3.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (first 70 terms from Vincenzo Librandi)

Crossrefs

Cf. A000009.

Programs

Maple

g:=x^3*product(1+x^j,j=1..60)/(1+x^3): gser:=series(g,x=0,57): seq(coeff(gser,x,n),n=1..54); # Emeric Deutsch, Apr 17 2006

Mathematica

Table[Count[Select[IntegerPartitions[n],Length[Union[#]] == Length[#] &], ?(MemberQ[#, 3] &)], {n, 60}] (* _Harvey P. Dale, Aug 19 2011 *)
nmax = 100; Rest[CoefficientList[Series[x^3/(1 + x^3) * Product[1 + x^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 30 2015 *)

Formula

G.f.: (x^3/(1 + x^3)) * Product_{j >= 1} (1 + x^j). - Emeric Deutsch, Apr 17 2006

Corresponding g.f. for "number of k's" is (x^k/(1 + x^k)) * Product_{j >= 1} (1 + x^j). - Joerg Arndt, Feb 20 2014

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

Extensions

Example clarified by Harvey P. Dale, Aug 19 2011