Pensieve header: A riddle via Jacob Tsimerman from https://fivethirtyeight.com/features/can-you-make-24/

T[n_]:=T[n,0];
T[0,j_]=1;
T[n_,0]:=1+Sum[T[k-1,n-k],{k,n}];
T[n_,j_]:=T[n,j]=1+Sum[T[k-1,n-k+j],{k,n}]+j T[n-1,j];

T[0]

Table[T[n],{n,20}]

Table[T[n],{n,20}] // FindSequenceFunction

Ts[n_]:=Sum[(((n-k+1))^k),{k,0,n}]

Table[Ts[n], {n,1,10}]