Editing Theta function (section)

=== Overpartition number sequence ===

The [[Maclaurin series]] for the reciprocal of the function {{math|''{{not a typo|ϑ}}<sub>01</sub>''}} has the numbers of [[Partition function (number theory)|over partition sequence]] as coefficients with a positive sign:<ref>{{cite journal | last=Mahlburg | first=Karl | title=The overpartition function modulo small powers of 2 | journal=Discrete Mathematics | volume=286 | issue=3 | date=2004 | doi=10.1016/j.disc.2004.03.014 | pages=263–267}}</ref>

: <math>\frac{1}{\theta_{4}(x)} = \prod_{n=1}^{\infty} \frac{1 + x^{n}}{1 - x^{n }} = \sum_{k=0}^{\infty} \overline{P}(k)x^{k}</math>
: <math>\frac{1}{\theta_{4}(x)} = 1+2x+4x^2+8x^3+14x^4+24x^5+40x^6+64x^7+100x^ 8+154x^9+232x^{10} + \dots</math>

If, for a given number <math>k</math>, all partitions are set up in such a way that the summand size never increases, and all those summands that do not have a summand of the same size to the left of themselves can be marked for each partition of this type, then it will be the resulting number<ref>{{cite journal|language=en|title=Elsevier Enhanced Reader|journal=Discrete Mathematics |date=28 April 2009 |volume=309 |issue=8 |pages=2528–2532 |doi=10.1016/j.disc.2008.05.007 |last1=Kim |first1=Byungchan |doi-access=free }}<!-- auto-translated by Module:CS1 translator --></ref> of the marked partitions depending on <math>k</math> by the overpartition function <math>\overline{P}(k)</math> .

First example:

: <math>\overline{P}(4) = 14</math>

These 14 possibilities of partition markings exist for the sum 4:

{| class="wikitable"
|(4), ('''4'''), (3+1), ('''3'''+1), (3+'''1'''), ('''3'''+'''1'''), (2+2), ('''2'''+2), (2+1+1), ('''2'''+1+1), (2+'''1'''+1), ('''2'''+'''1'''+1), (1+1+1+1), ('''1'''+1+1+1)
|}

Second example:

: <math>\overline{P}(5) = 24</math>

These 24 possibilities of partition markings exist for the sum 5:

{| class="wikitable"
|(5), ('''5'''), (4+1), ('''4'''+1), (4+'''1'''), ('''4'''+'''1'''), (3+2), ('''3'''+2), (3+'''2'''), ('''3'''+'''2'''), (3+1+1), ('''3'''+1+1), (3+'''1'''+1), ('''3'''+'''1'''+1), (2+2+1), ('''2'''+2+1), (2+2+'''1'''), ('''2'''+2+'''1'''),
(2+1+1+1), ('''2'''+1+1+1), (2+'''1'''+1+1), ('''2'''+'''1'''+1+1), (1+1+1+1+1), ('''1'''+1+1+1+1)
|}