What should I do to get my name in the koseki

I had a Japanese baby with a Japanese girl, I wanna get my name as a father in the koseki but I don’t wanna marry the girl. What should I do to get my name in the koseki

You got some answer from reddit(is it you?)

anyway ... this is the way..bring you child mother along

Assuming your partner doesn't have a husband, just go to city hall and fill out the affiliation report (認知届).

related link on how to
http://marriage.visaemon.jp/article/14342159.html

check your city hall for procedure and such. below is example
https://www.city.amagasaki.hyogo.jp/kurashi/todokede/koseki/066ninchi....

this is the form you need (認知届) ninchi-todoke.
https://www.mofa.go.jp/mofaj/toko/todoke/koseki/pdfs/15ninti.pdf