Of course it would be better to have patronymic profiles but I didn't find any.
The suggested LNAB profiles do have information on them.
http://www.wikitree.com/wiki/Kool-248
http://www.wikitree.com/wiki/Kool-247
I think this is a mistake though
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Jacob Barentsen KOOL, born abt 1638, baptised 25 Sep 1639 at the Dutch Reformed Church, New Amsterdam, New Netherland, wed Maritje Symonse in 1660, died 1719 in NY;
It should probably be baptised before 25 Sep 1639, as that's the date of the first record in New Amsterdam and it's not for this baptism.
Barent:
I posted his baptism records on his profile, both the PPP and the suggested.
He was in Nieuw-Nederland by 1640.
* 1640 23 Sep Aeltje, Barent Jacobz. Wit.: Jan Hermanszen, Leuntie Alberts.
He definitely did not die before 1657 as that's when he baptized his last child.
* 1657 29 Aug Pieter, Barent Jacobszen, Marrietie Leenderts. Wit.: Jan Corneliszen, Adriaentie Trommels.
Barent definitely used Cool.
* 1655 10 Oct Arent, Barent Jacobszen Cool, Marrietje Leenderts. Wit.: Jan Prie, Aechtje Leenderts.
He also appeared in 1645 testifying that he arrived in 1638 aboard Den Dolphyn and signed it Barent Jacobsz Cool. He also says that he is 34. p. 167.
http://www.newnetherlandinstitute.org/files/5813/5543/8913/RegisterofPSVol2.pdf
Barent is listed in the NYGBR 141.1 in Small Burghers of New Amsterdam dated 1657 April 13.14.
He supposedly died in 1676 in Kingston.
Jacob:
There is no record of him in Nieuw-Nederland, he is not in the NYGBR or listed as an ancestor anywhere. The Holland Dames lists Kool, Barent Jacobsen as the eligible ancestor.
Cool:
The Kool line is not backwards projected though it does belong to a family that was determined to not belong to Jacob. A Kool line, though again not Jacobs, has been traced back to an Adriaen Kool 1450. My understanding is that it's on file with the NYGBS.
I haven't looked at all of Barent's children's baptisms but his son Jacob appears with both patronymic and C/Kool in the Kingston book https://archive.org/stream/baptismalmarriag00king#page/4/mode/2up Leendert as well.
I'll look to see if I can't find microfilm of Jacob's baptism that Albertus has listed but admittedly I don't have the hang of the Amsterdam books yet. Oh they're pretty legible at least!