A007318 - Pascal's triangle and its powers.

[Antti Karttunen]

Cher(e)s Fanaticien(ne)s,

just realized that when the Pascal's triangle is considered as
an infinite lower triangular matrix, e.g. as

(1 0 0 0 0 0 0 0 0 ...)
(1 1 0 0 0 0 0 0 0 ...)
(1 2 1 0 0 0 0 0 0 ...)
(1 3 3 1 0 0 0 0 0 ...)
(1 4 6 4 1 0 0 0 0 ...)

then its nth power (n in Z, thus including also n=0, the identity
matrix,
and n=-1, A007318's inverse) seems to be given by the recursive
construction
T(0,0) = 1,
T(n,k) = T(n-1,k-1) + n*T(n-1,k)

(e.g. see A007318's "square" A038207 and the "cube" A027465).


What about any other triangles generated with such simple recursive
rules,
e.g. the Delannoy numbers A008288 (T(n,k) = T(n-1,k-1) + T(n-1,k) +
T(n-2,k-1))
or the Catalan's triangle A009766 (T(n,k) = T(n,k-1) + T(n-1,k), T(n,0)
= 1)
do any of these have such "meta-rules" ???


Terveisin,

Antti Karttunen