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

0,5

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]

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.

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A029049 A094983 A238218 * A118301 A018121 A256636

Adjacent sequences: A015741 A015742 A015743 * A015745 A015746 A015747

Keyword

nonn

Author

Clark Kimberling

Extensions

Corrected and extended by Dean Hickerson, Oct 10, 2001

Status

approved