ClearAll[Sum1];
Attributes[Sum1] = {HoldFirst};
Sum1[f_, {ks_List, ksummax_Integer?Positive}] :=
  With[{perms =
     Flatten[Permutations /@
         IntegerPartitions[#, {Length@ks}, Range[0, #]] & /@
       Range[2, ksummax], 2]}, Total[f /. (Thread[ks -> #] & /@ perms)]];

[CapitalPhi][dmax_, ksummax_][n_, s_,
  t_List] := (1/6*
    Sum[t[[i + 1]] t[[j + 1]] t[[n - i - j + 1]], {i, 0, n}, {j, 0,
      n - i}] +
   Sum1[Sum[
     No[d][Sequence @@ Table[k[i], {i, 2, n}]]*
      Product[t[[i + 1]]^k[i]/k[i]!, {i, 2, n}]*E^(d t[[2]])*s^d, {d,
      0, dmax}], {k[#] & /@ Range[2, n], ksummax}])

In[218]:= [CapitalPhi][2, 4][3, s, Array[t, 4, 0]] // Simplify

Out[218]= 1/24 (4 t[1]^3 + 24 t[0] t[1] t[2] + 12 t[0]^2 t[3] +
   12 t[3]^2 No[0][0, 2] + 4 t[3]^3 No[0][0, 3] +
   t[3]^4 No[0][0, 4] + 24 t[2] t[3] No[0][1, 1] +
   12 t[2] t[3]^2 No[0][1, 2] + 4 t[2] t[3]^3 No[0][1, 3] +
   12 t[2]^2 No[0][2, 0] + 12 t[2]^2 t[3] No[0][2, 1] +
   6 t[2]^2 t[3]^2 No[0][2, 2] + 4 t[2]^3 No[0][3, 0] +
   4 t[2]^3 t[3] No[0][3, 1] + t[2]^4 No[0][4, 0] +
   12 E^t[1] s t[3]^2 No[1][0, 2] + 4 E^t[1] s t[3]^3 No[1][0, 3] +
   E^t[1] s t[3]^4 No[1][0, 4] + 24 E^t[1] s t[2] t[3] No[1][1, 1] +
   12 E^t[1] s t[2] t[3]^2 No[1][1, 2] +
   4 E^t[1] s t[2] t[3]^3 No[1][1, 3] +
   12 E^t[1] s t[2]^2 No[1][2, 0] +
   12 E^t[1] s t[2]^2 t[3] No[1][2, 1] +
   6 E^t[1] s t[2]^2 t[3]^2 No[1][2, 2] +
   4 E^t[1] s t[2]^3 No[1][3, 0] +
   4 E^t[1] s t[2]^3 t[3] No[1][3, 1] + E^t[1] s t[2]^4 No[1][4, 0] +
   12 E^(2 t[1]) s^2 t[3]^2 No[2][0, 2] +
   4 E^(2 t[1]) s^2 t[3]^3 No[2][0, 3] +
   E^(2 t[1]) s^2 t[3]^4 No[2][0, 4] +
   24 E^(2 t[1]) s^2 t[2] t[3] No[2][1, 1] +
   12 E^(2 t[1]) s^2 t[2] t[3]^2 No[2][1, 2] +
   4 E^(2 t[1]) s^2 t[2] t[3]^3 No[2][1, 3] +
   12 E^(2 t[1]) s^2 t[2]^2 No[2][2, 0] +
   12 E^(2 t[1]) s^2 t[2]^2 t[3] No[2][2, 1] +
   6 E^(2 t[1]) s^2 t[2]^2 t[3]^2 No[2][2, 2] +
   4 E^(2 t[1]) s^2 t[2]^3 No[2][3, 0] +
   4 E^(2 t[1]) s^2 t[2]^3 t[3] No[2][3, 1] +
   E^(2 t[1]) s^2 t[2]^4 No[2][4, 0])

LHS = 2 Composition[D[#, t[1]] &, D[#, t[2]] &,
       D[#, t[3]] &] @[CapitalPhi][2, 4][3, s, Array[t, 4, 0]] //
    Simplify // Expand;
RHS = Composition[D[#, t[1]] &, D[#, t[1]] &, D[#, t[1]] &,
       D[#, t[2] ] &, D[#, t[2]] &,
       D[#, t[2] ] &]@[CapitalPhi][3, 6][3, s, Array[t, 4, 0]] -
     Composition[D[#, t[1] ] &, D[#, t[1] ] &,
        D[#, t[2] ] &]@[CapitalPhi][2, 4][3, s, Array[t, 4, 0]]*
      Composition[D[#, t[1] ] &, D[#, t[2] ] &,
        D[#, t[2] ] &]@[CapitalPhi][2, 4][3, s, Array[t, 4, 0]] //
    Simplify // Expand;

coeffLHS = Flatten@CoefficientList[LHS, {Sequence @@ Array[t, 4, 0], E^t[1], s}];
coeffRHS = Flatten@CoefficientList[RHS, {Sequence @@ Array[t, 4, 0], E^t[1], s}];

lim = Min[Length@coeffLHS, Length@coeffRHS];
varsLHS = Variables[coeffLHS[[1 ;; lim]]];
varsRHS = Variables[coeffRHS[[1 ;; lim]]];
Solve[LogicalExpand[coeffLHS[[1 ;; lim]] == coeffRHS[[1 ;; lim]]],
 Union[varsLHS, varsRHS]]