A015744 Number of partitions of n into distinct parts, none being 2.

1, 1, 0, 1, 2, 2, 2, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 22, 27, 32, 37, 44, 52, 60, 70, 82, 95, 110, 127, 146, 169, 194, 221, 254, 291, 331, 377, 429, 487, 553, 626, 707, 800, 903, 1016, 1145, 1288, 1445, 1622, 1819, 2036, 2278, 2546, 2842, 3172, 3536, 3936, 4381

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

With offset 2 (and a(0)=a(1)=0) the number of 2's in all partitions of n into distinct parts. [Joerg Arndt, Feb 20 2014]

Examples

			a(8)=4 because we have [8],[7,1],[5,3] and [4,3,1].
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Cristina Ballantine, Mircea Merca, On identities of Watson type, Ars Mathematica Contemporanea (2019) Vol. 17, 277-290.

Crossrefs

Cf. A025147, A015745, A015746, A015750, A015753, A015754, A015755.

Programs

Maple

g:=(1+x)*product(1+x^j,j=3..80): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..57); # Emeric Deutsch, Apr 09 2006

Mathematica

CoefficientList[Series[Product[1+q^n, {n, 1, 60}]/(1+q^2), {q, 0, 60}], q]
Table[Count[Select[IntegerPartitions[n], DeleteDuplicates[#] == # &], x_ /; ! MemberQ[x, 2]], {n, 0, 57}] (* Robert Price, May 17 2020 *)

Formula

G.f.: (1+x)*product(j>=3, 1+x^j ). - Emeric Deutsch, Apr 09 2006

a(n+2)=sum_{k=1..floor(n/2)} (-1)^(k-1)*A000009(n-2*k). - Mircea Merca, Feb 20 2014

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

Extensions

Corrected and extended by Dean Hickerson, Oct 10 2001