A100522 Number of partitions of n into parts free of both odd squares and even numbers which are not squares, the odd parts they occur with a single multiplicity, there is no restriction on the even parts.

1, 0, 0, 1, 1, 1, 0, 2, 2, 1, 1, 3, 3, 2, 2, 5, 6, 3, 5, 8, 9, 7, 8, 13, 14, 10, 14, 19, 20, 17, 20, 29, 30, 26, 32, 42, 45, 41, 47, 63, 64, 60, 70, 88, 91, 87, 99, 124, 128, 123, 143, 172, 179, 176, 200, 240, 246, 246, 279, 325, 337, 338, 381, 440, 456, 461, 519, 590, 615

Offset

0,8

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

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

Formula

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

Example

a(16)=6 because 16 = 13+3 = 11+5 = 7+5+4 = 5+3+4+4 = 4+4+4+4.

Maple

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

Mathematica

With[{m=80}, CoefficientList[Series[Product[(1+x^(2*k-1))/(1-(-1)^k *x^(k^2)), {k, m+2}], {x, 0, m}], x]] (* G. C. Greubel, Mar 28 2023 *)

Prog

(Magma)
m:=80;
f:= func< x | (&*[(1+x^(2*k-1))/(1-(-1)^k*x^(k^2)): k in [1..m+2]]) >;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( f(x) )); // G. C. Greubel, Mar 28 2023
(SageMath)
m=80
def f(x): return product( (1+x^(2*k-1))/(1-(-1)^k*x^(k^2)) for k in range(1, m+2))
def A100522_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( f(x) ).list()
A100522_list(m) # G. C. Greubel, Mar 28 2023

Crossrefs

Cf. A100926.

Sequence in context: A348890 A110659 A308068 * A258140 A140408 A047080

Adjacent sequences: A100519 A100520 A100521 * A100523 A100524 A100525

Keyword

nonn

Author

Noureddine Chair, Nov 25 2004

Extensions

Offset corrected by G. C. Greubel, Mar 28 2023

Status

approved