Column[Table[
Binomial[n, k],
{n, 0, 6}, {k, 0, n}], Center]
Column[Table[
Binomial[r, c] +
Binomial[r + 1, c] +
Binomial[r + 1, c + 1],
{r, 0, 5}, {c, 0, r}], Center]
Column[Table[
Binomial[r, c] + Binomial[r + 1, c] + Binomial[r + 1, c + 1] +
Binomial[r + 1, c] + Binomial[r + 2, c] + Binomial[r + 2, c + 1] +
Binomial[r + 1, c + 1] + Binomial[r + 2, c + 1] + Binomial[r + 2, c + 2],
{r, 0, 4}, {c, 0, r}], Center]
Column[Table[
(Binomial[r, c] + Binomial[r + 1, c] + Binomial[r + 1, c + 1] +
Binomial[r + 1, c] + Binomial[r + 2, c] + Binomial[r + 2, c + 1] +
Binomial[r + 1, c + 1] + Binomial[r + 2, c + 1] + Binomial[r + 2, c + 2]) +
(Binomial[r + 1, c] + Binomial[r + 2, c] + Binomial[r + 2, c + 1] +
Binomial[r + 2, c] + Binomial[r + 3, c] + Binomial[r + 3, c + 1] +
Binomial[r + 2, c + 1] + Binomial[r + 3, c + 1] + Binomial[r + 3, c + 2]) +
(Binomial[r + 1, c + 1] + Binomial[r + 2, c + 1] + Binomial[r + 2, c + 2] +
Binomial[r + 2, c + 1] + Binomial[r + 3, c + 1] + Binomial[r + 3, c + 2] +
Binomial[r + 2, c + 2] + Binomial[r + 3, c + 2] + Binomial[r + 3, c + 3]),
{r, 0, 3}, {c, 0, r}], Center]
From that you may be able to see the pattern for how to generate all subsequent layers.