A100926 Number of partitions of n into parts free of odd squares and the only number with multiplicity in the unrestricted partitions is the number 2.

1, 0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 23, 27, 33, 40, 48, 57, 69, 81, 97, 113, 134, 157, 184, 214, 250, 290, 337, 389, 451, 519, 598, 688, 789, 904, 1035, 1181, 1348, 1535, 1746, 1983, 2250, 2549, 2885, 3261, 3682, 4154, 4680, 5268, 5923, 6656, 7468

Offset

1,5

Comments

This is also the inverted graded generating function for the number of partitions in which no square parts are present

Links

Table of n, a(n) for n=1..56.

Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.

James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.

Formula

G.f.: Product_{k>=0}(1+x^k)/(1-(-1)^k*x^(k^2)).

Example

a(10)=8 because 10 =8+2 =7+3 =6+4 =5+3+2 =6+2+2 =4+2+2+2 =2+2+2+2+2.

Maple

series(product((1+x^k)/(1-(-1)^k*x^(k^2)), k=1..100), x=0, 100);

Mathematica

terms = 56; Product[(1 + x^k)/(1 - (-1)^k*x^(k^2)), {k, 1, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Dec 14 2018 *)

Crossrefs

Sequence in context: A117409 A092833 A280664 * A351008 A258875 A179241

Adjacent sequences: A100923 A100924 A100925 * A100927 A100928 A100929

Keyword

nonn

Author

Noureddine Chair, Nov 22 2004

Status

approved