Barnabas Sedgwick  [parents unknown]
b. est 1810, Pennsylvania2
d. bef 18502, [Louisville, Jefferson, Kentucky]
m. September 2, 1834, Cincinatti, Hamilton, Ohio, Lucinda Duncan3 certificate

Lucinda Duncan  [parents unknown]
b. abt 1808, Virginia2
d. June 1872, [Louisville, Jefferson, Kentucky]4
bu June 10, 1872, Louisville, Jefferson, Kentucky, Eastern Cemetery4

Census records:

None found for Barnabas
Widow Lucinda:
1850 KY Jefferson, Louisville
1860 KY Jefferson, Louisville
1870 KY Jefferson, Louisville

We do not know who Barnabas' parents were, but there is a clue. Thomas Stewart Sedwick, who served with Barnabas' son Thomas Duncan Sedgwick during the Civil War, said this in a letter for which I have a partial (undated) transcription:

"I remember being told about a Thomas Sedgwick who went into Pennsylvania and not having been heard from was thought to have gone to Kentucky. A Sedwick was an officer in the Miliitia in Ky, grandfather to the one I served with in the Army (114th U.S.C.T.) This Thomas must surely be the grandfather of P.K.S. [Peter Kiner Sedwick] who wrote this letter and probably lost in Kentucky."
Daughter Ione Sedgwick said in censuses of 1880 and 1900 that her father was born in Pennsylvania.

Given that Barnabas and Lucinda were married in Cincinatti, Hamilton, Ohio, I have wondered if this Thomas Sedgewick is his father:

1817 OH Hamilton, Cincinatti
1820 OH Hamilton, Cincinatti

Children:

Thomas Duncan Sedgwick
  b. abt 1837, [Louisville, Jefferson?], Kentucky2
  d. April 26, 1879 [Washington, DC?]6a
  m1 June 5, 1861, Floyd County, Indiana, Priscilla Tuley5
  m2 September 3, 1875, Washington, DC, Emma L Unknown6a
  served in the Civil War, Union Army
Ione Sedgwick
  b. Feb 18391a,2
  d. July 29, 1914, Aberdeen, Chehalis, Washington13
  m. Moses Hart1a

A grand-daughter of Ione wrote that Ione claimed another brother Charles who served in the Civil War Confederate Army, but no record of him has been found. There was a Charles Zadock Sedwick who fought in the Confederate Cavalry; perhaps she heard some mention of him and the connection became distorted.

Sources and notes: (all sources for this project)

1. Records of descendants/researchers of this family
a. Chuck Downen, descendant of Ione Sedgwick and Moses Hart

2. Census records, any of those indicated.

3. Ohio, Hamilton County Probate Court Marriage Records
a. Vol 4 (Restored), includes 1834, filmed on LDS FHL US/CAN Film 344452 image here

4. Eastern Cemetery Records, Louisville, Kentucky
compiled for the Louisville Genealogical Society, c1994
Call Number 976.944/L1 V38L v. 2, available at LDS FHC Salt Lake City
this record:
Date  Name  Range  Grave  Section  Book  Page  
June 10, 1872  Lucinda Sedwich  82312D00454

5. Floyd County, Indiana Index to Marriage Records R-Z Book 1 Vol 4 1845-1920
found at GenealogyLibrary.com, also available LDS Film 1266794 Items 6-7
these records:
Groom/BrideSpouseDate of MarriageBook/Page
Sedgewick Thomas DPricilla TuleyJune 5 18615/259
Sedgewick RoseJames BruckerNov 12 189110/72

6. Military Service or Pension Records
a. Pension Records of Thomas Duncan Sedgwick, complete file obtained by Chuck Downen from NARA

13. Washington State Archives - Digital Archives
this record: http://www.digitalarchives.wa.gov/Record/View/BDB632825C3AD81CED4AC3390A9C86C4
  Da Reference Number: {CA30F5D3-8DF4-4225-959C-26546D97C1B0}
  Image Number: 712
  Document Number: 99
  Document Reference Id: 221
  Name: Ione Sedgwick Hart
  Date Of Death: 29 Jul 1914
  Age: 75
  Gender: Female
  Father Name: Sedgwick
  Mother Name: Knaught
  Batch Id: 275617
  Batch Locality: Washington, United States
  Death Place: Aberdeen, Chehalis, Washington
  Mother Name Surname: Knaught
I don't have any idea why her mother's surname would be listed as Knaught.

Descendants / Researchers:

Chuck Downen <cjdownen@charter.net>
descendant of Ione Sedgwick and Moses Hart
Questions? Have more data? genealogy@sedgwick.org