A088857 - OEIS

A088857

Let Product[1+Sum[b(i,j) x^(i*j),{i,1,Infinity}],{j,1,Infinity}]=1+Sum[c(n) x^n,{n,1,Infinity}], where b(i,j) is plus or minus one and c(n) is plus or minus one or zero. Furthermore, let b(1,1)=1 (for definiteness). Then, for a given n, a(n) is the number of ways in which the coefficients b(i,j) i<=n, j<=n can be chosen.

1, 2, 6, 30, 80, 634, 1424, 5392, 10677, 38052, 39051, 815616, 1421550

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

In all of these series, the coefficients b(n) will be congruent, modulo 2, to the number of unrestricted partitions of n. Not much is known about the parity of the partition function, although much is known if the modulus 2 is replaced by other numbers such as 5 or 7. As for the sequence proposed here, it is not even known if it is finite or infinite. I have found coefficients satisfying the given conditions for n as large as 70.

REFERENCES

G. E. Andrews, The Theory of Partitions.

LINKS

Table of n, a(n) for n=1..13.

EXAMPLE

a(2)=2 because (1+x+x^2)(1-x^2)=1+x+0x^2+... and (1+x-x^2)(1+x^2)=1+x+0x^2+... But, (1+x+x^2)(1+x^2)=1+x+2x^2+... and (1+x-x^2)(1-x^2)=1+x-2x^2+...

CROSSREFS

Sequence in context: A117849 A328417 A290760 * A099081 A051844 A335396

Adjacent sequences: A088854 A088855 A088856 * A088858 A088859 A088860

KEYWORD

hard,nonn

AUTHOR

David S. Newman, Nov 29 2003

STATUS

approved