adc[expr_] := Simplify[
D[expr, s]*2p+D[expr, p]*s*p
];
adc[0, expr_] := expr;
adc[1, expr_] := adc[expr];
adc[n_, expr_] /; n>1 := adc[adc[n-1, expr]];

adc[1,s]

adc[1,p]

adc[2,p]

adc[3,p]

adc[4,p]

adc[10,p]

Table[
Expand[adc[n,p] /. {s -> a+b, p -> a b}],
{n,0,9}
]

Table[2^(n+1)-n-2, {n,0,9}]

Expand[adc[10,p] /. {s -> a+b, p -> a b}]

See http://oeis.org/wiki/Eulerian_numbers,_triangle_of !

Expand[adc[7,p] /. {s -> a+b, p -> a b}] /. {a->1, b->1}

Expand[CoefficientList[Series[t Exp[x t]/(E^t-1), {t, 0,10}], t]*Table[k!, {k,0,10}]]

Series[2 Exp[x t]/(E^t+1), {t, 0,10}]

Expand[10!(-((31 x)/725760)+((17 (x^3))/241920)-((x^5)/28800)+((x^7)/120960)-((x^9)/725760)+((x^10)/3628800))]

Expand[CoefficientList[Series[2 Exp[x t]/(E^t+1), {t, 0,10}], t]*Table[(k+1)!, {k,0,10}]]

adc[10,s]

Simplify[adc[10,s] /. {s -> a+b, p -> a b}]

DSolve[
{D[s[t],t]==2p[t], D[p[t], t]==s[t]p[t]},
{s[t], p[t]},
t
]

DSolve[
{
D[g[t,s,p], t]==2p D[g[t,s,p], s]+s p D[g[t,s,p], p],
g[0,s,p]==s
},
{g[t,s,p]},
{t, s, p}
]

gf=Sum[t^n/n! adc[n,s], {n, 0, 10}] + O[t]^11

Simplify[gf*(1-E^(-s t)) /. {s->a+b, p -> a b}]

DSolve[
{
D[g[t,a,b], t]==a b  D[g[t,a,b], a]+a b  D[g[t,a,b], b],
g[0,a,b]==a
},
{g[t,a,b]},
{t,a,b}
]

DSolve[
{
D[g[t,a,b], t]==b D[g[t,a,b], a]+a  D[g[t,a,b], b],
g[0,a,b]==a
},
g[t,a,b],
{t,a,b}
]