probably New York,  New York Ward 9 District 14, middle initial may be E from some other records

not listed in the 1870 census

 Probably New York City, NY, Ward 9, District 14

alive and living with grandmother Elizabeth Cornell in 1880 census

Manhattan birth certificate # 62558, listed as being child #4, place of birth 169 Charles St. Manhattan, NY, mother Mary Frances Cole, born , NY, age 24, father Garret Servans Cole, carpenter, age 30, born, death record  and gravestone showed date of birth as being 1872

Birth place for both was listed as being New York, New York on birth certificate for child #6, Nelson F. Cole

by Laurel Grove Cemetery, New Jersey headstone, section 8, birth date shows as being 1872

1910 census records list date of birth as Feb. 1872

New York Births and Christenings, 1640-1962 for Garret Serrvans Cole
Name:     Mary Elizabeth Cole
Gender:     Female
Birth Date:     16 Feb 1871
Birthplace:     Manhattan, New York, New York, USA
Father's Name:     Garret Serrvans Cole
Mother's Name:     Mary Frances Perry
Indexing Project (Batch) Number:     C59461-8
System Origin:     New_York-ODM
Source Film Number:     1322035

Death record 16236, New Jersey, James Young was the registrar of the death, by Laurel Grove Cemetery, New Jersey headstone, section 8, lot 24, grave 2, place of death is listed as Paterson on burial death records, buried with son and daughter -in-law at Laurel Grove

grave records indicate widow of George, undertaker Robert S. Browning, owner of lot Mary E. Fischer (6 X 16), deed 2823

New York Births and Christenings, 1640-1962 for Cole
Name:     Cole
Gender:     Female
Birth Date:     28 Jul 1876
Birthplace:     BIRTH CERTIFICATES, MANHATTAN, NEW YORK, NEW YORK
Father's Name:     Garrett S. Cole
Mother's Name:     Mary Frances Peovy
Indexing Project (Batch) Number:     C71505-4
System Origin:     New_York-ODM
Source Film Number:     1322104

alive by 1880 census

IGI record C715111, SC 1322155 and Manhattan birth record and father's military pension records

New York Births and Christenings, 1640-1962 for Nelson F. Cole
Name:     Nelson F. Cole
Gender:     Male
Birth Date:     10 Aug 1880
Birthplace:     BIRTH CERTIFICATES, MANHATTAN, NEW YORK, NEW YORK
Father's Name:     Garrett S. Cole
Mother's Name:     Mary Francis Perry Cole
Indexing Project (Batch) Number:     C71511-1
System Origin:     New_York-ODM
Source Film Number:     1322155

dead by october 18, 1918 from military pension records of mother

by father's pension records, probably Paterson, New Jersey as father  a resident by 1883

Birth date listed as Oct. 1883 by 1900 census, state listed there