Archambaud VIII of Bourbon
Archambaud VIII of Bourbon, nicknamed "the Great", (1189–1242) was a ruler ("sire") of Bourbonnais in the modern region of Auvergne, France.
His parents were Guy II of Dampierre and Mathilde of Bourbon.
Archambaud’s first wife was Alix de Forez.
They married in 1205.
Before she was repudiated, Alix bore:
Archambaud later married Beatrice de Montluçon, who bore him;
Peter II, Duke of Bourbon
Peter II, Duke of Bourbon (1 December 1438 – 10 October 1503 in Moulins), was the son of Charles I, Duke of Bourbon, and Agnes of Burgundy, and a member of the House of Bourbon.
He and his wife Anne of France ruled as regents during the minority of Charles VIII of France.
Archambaud IX of Bourbon
Archambaud IX of Bourbon (died 15 January 1249), called Le Jeune ("The Young"), was a ruler (sire) of Bourbonnais in the modern region of Auvergne, France.
He was the son of Archambaud VIII of Bourbon.
He married Yolande I, Countess of Nevers.
They had:
He died in Cyprus on 15 January 1249 during the Seventh Crusade.
Kaya Alp
Kaya Alp was, according to Ottoman tradition, the son of Kızıl Buğa and the father of Suleyman Shah, who was, in turn, the grandfather of Ertuğrul, and the great grandfather of the Ottoman Empire founder, Osman I.
Beatrice of Bourbon, Queen of Bohemia
Beatrice of Bourbon( 1320 – 23 December 1383) was a French noblewoman.
A member of the House of Bourbon, she was by marriage Queen of Bohemia and Countess of Luxembourg.
She was the youngest daughter of Louis I, Duke of Bourbon, and Mary of Avesnes.
Agnes of Dampierre
Agnes of Dampierre( 1237 – 7 September 1288) was Lady of Bourbon and heiress of all Bourbon estates.
She was the daughter of Archambaud IX de Dampierre and Yolande I, Countess of Nevers.
She married John of Burgundy, the son of Hugh IV, Duke of Burgundy.
Her daughter, Beatrix of Bourbon, married Robert, Count of Clermont in 1272 and their eldest son Louis I," le Boiteux" became the first Duke of Bourbon.
Mathilde of Bourbon
Mathilde of Bourbon( – 18 June 1228) was a French noblewoman who was the ruling Lady of Bourbon from 1171 until her death.
Matilda II, Countess of Nevers
Matilda II, Countess of Nevers( 1234/35–1262), also known as Maud of Dampierre or Mathilda II of Bourbon, was a sovereign Countess of Nevers, Countess of Auxerre, Countess of Tonnerre and Lady of Bourbon.
She was a daughter of Archambaud IX of Bourbon and Yolande de Châtillon, Countess of Nevers.
As heiress to the counties of Nevers, Auxerre and Tonnerre, she was married off to Odo, Count of Nevers, the son and heir of Hugh IV, Duke of Burgundy.
By this marriage Odo reunited two important areas that had previously been united as the Duchy of Burgundy, but he died leaving no male heirs, and so the united territory passed to his brother Robert II, Duke of Burgundy.
After her father's death Matilda also became Lady of Bourbon.
With Odo she had three children:
Princess Margaret of Denmark
Princess Margaret of Denmark
( Margrethe Françoise Louise Marie Helene; 17 September 1895 – 18 September 1992) was a Danish princess by birth and a princess of Bourbon- Parma as the wife of Prince René of Bourbon- Parma.
She was the youngest grandchild of Christian IX of Denmark and Queen Louise.
John of Burgundy (1231–68)
John of Burgundy( 1231 – 29 September 1268) was a Count of Charolais and Lord of Bourbon.
He was a younger son of Duke Hugh IV of Burgundy and his wife, Yolande of Dreux.
John married in February 1248 to Agnes( d. 1288), the heiress of Lord Archambaud IX of Bourbon from the House of Dampierre.
After the death of his father- in law in 1249 John became Lord of Bourbon in right of his wife.
John and Agnes had a daughter, Beatrice( d. 1310), who inherited possessions from both of her parents.
In 1272, she married the royal Prince Robert, Count of Clermont and thereby founded the Capetian dynasty of the Bourbons.

Question: Who is the paternal grandfather of Archambaud Ix Of Bourbon?
Answer: Guy II of Dampierre