Merci Jandri !
Effectivement, on retrouve très rapidement plusieurs $P_q$ donnés dans ce fil en cravachant.
> with(PolynomialTools); factor(CoefficientList(convert(series(((exp(t)-1)/t)^p, t, 14), polynom), t)) ;
[1, (1/2)*p, (1/24)*p*(1+3*p), (1/48)*p^2*(p+1), (1/5760)*p*(15*p^3+30*p^2+5*p-2), (1/11520)*p^2*(p+1)*(3*p^2+7*p-2),
(1/2903040)*p*(63*p^5+315*p^4+315*p^3-91*p^2-42*p+16), (1/5806080)*p^2*(p+1)*(9*p^4+54*p^3+51*p^2-58*p+16),
(1/1393459200)*p*(135*p^7+1260*p^6+3150*p^5+840*p^4-2345*p^3+540*p^2+404*p-144),
(1/2786918400)*p^2*(p+1)*(15*p^6+165*p^5+465*p^4-17*p^3-648*p^2+548*p-144),
(1/367873228800)*p*(99*p^9+1485*p^8+6930*p^7+8778*p^6-8085*p^5-8195*p^4+11792*p^3-2068*p^2-2288*p+768),
(1/735746457600)*p^2*(p+1)*(9*p^8+156*p^7+834*p^6+1080*p^5-1927*p^4-1252*p^3+4156*p^2-3056*p+768),
(1/24103053950976000)*p*(12285*p^11+270270*p^10+2027025*p^9+5495490*p^8+315315*p^7-12882870*p^6
+5760755*p^5+14444430*p^4-15875860*p^3+2037672*p^2+3327584*p-1061376),
(1/48206107901952000)*p^2*(p+1)*(945*p^10+23625*p^9+201600*p^8+609210*p^7-113715*p^6-2207175*p^5
+1817786*p^4+3161188*p^3-6544568*p^2+4388960*p-1061376)]
