A100926 -id:A100926 - cp's OEIS frontend

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

Noureddine Chair, Nov 25 2004

nonn

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

G. C. Greubel, Table of n, a(n) for n = 0..5000
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011, 2004.

Cf. A100926.

m:=80;
f:= func< x | (&*[(1+x^(2*k-1))/(1-(-1)^k*x^(k^2)): k in [1..m+2]]) >;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( f(x) )); // G. C. Greubel, Mar 28 2023

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

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 *)

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. = PowerSeriesRing(ZZ, prec)
    return P( f(x) ).list()
A100522_list(m) # G. C. Greubel, Mar 28 2023

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

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

A100989 Number of partitions of n into parts free of odd hexagonal numbers and the only number with multiplicity in the unrestricted partitions is the number 2 with multiplicity of the form 3k+l, where k is a positive integer and l=0,1.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 6, 6, 9, 11, 13, 16, 20, 20, 23, 29, 35, 41, 49, 59, 68, 82, 96, 112, 131, 154, 178, 207, 242, 277, 321, 371, 425, 489, 562, 641, 733, 839, 953, 1086, 1236, 1399, 1588, 1798, 2032, 2295, 2592, 2917, 3285

Offset: 1

Noureddine Chair, Nov 29 2004

nonn

			a(15)=20 because 15 =13+2 =12+3 =11+4 =10+5 =10+3+2 =9+6=9+4+2 =8+7 =8+5+2 =8+4+3 =7+6+2 =7+5+3 =6+5+4 =6+4+3+2 =9+2+2+2 =7+2+2+2+2 =6+3+2+2+2 =5+4+2+2+2 =4+3+2+2+2+2 =3+2+2+2+2+2+2"

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

Cf. A100926, A100927.

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

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

cp's OEIS Frontend

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.

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