Table[(D[(a+1)^n, {a, m}]/m!) /. a -> 0,{n, 0,7}, {m, 0, n}]

(x=1/(a+1);Table[CoefficientList[x*=(a+1),a],{7+1}])

p1[n_Integer?Positive, 1] := 1;
p1[n_Integer?Positive, n_Integer?Positive] := 1;
p1[n_Integer?Positive, m_Integer?Positive /; m < n] :=
p1[n-1, m] + p1[n-1, m-1];

Table[p1[n, m], {n, 1, 7 + 1}, {m, 1, n}]

Table[Binomial[n, m], {n, 0, 7}, {m, 0, n}]

Table[n!/(m!*(n - m)!), {n, 0, 7}, {m, 0, n}]

Table[(-1)^m*(Pochhammer[-n, m]/m!), {n, 0, 7}, {m, 0, n}]

Join[{{1}}, Table[List @@ Expand[(a + 1)^n] /. {a -> 1}, {n, 1, 7}]]

Table[CoefficientList[(a + 1)^n, a], {n, 0,7}]

NestList[Prepend[#1, 0] + Append[#1, 0] & , {1}, 7]

Table[(D[(a+1)^n, {a, m}]/m!) /. a -> 0, {n, 0, 7}, {m, 0, n}]

(x = 1/(a+1); Table[CoefficientList[x *= (a+1), a], {7+1}])

Daniel Lichtblau

f[1,1] = 1
f[1,n_Integer] /; n>1 := 1
f[n_Integer,n_] /; n>1 := 1
f[j_Integer,n_Integer] /; (n>1&&j>1&&j<n) := f[j,n] = f[j-1,n-1] +
f[j,n-1]

Table[ f[x,8],{x,1,8}]

Marten

NestList[Flatten[{1, ListConvolve[{1, 1}, #], 1}] &, {1}, 8]