OFFSET
1,3
FORMULA
A128132 * A007318 as infinite lower triangular matrices (assuming the top of the Pascal triangle A007318 is shifted from (0,0) to (1,1)).
From Petros Hadjicostas, Jul 26 2020: (Start)
T(n,k) = n*binomial(n-1, k-1) - binomial(n-2, k-1)*[n <> k] for 1 <= k <= n, where [ ] is the Iverson bracket.
Bivariate o.g.f.: x*y*(1 - x + x^2*(1 + y))/(1 - x*(1 + y))^2.
T(n,k) = T(n-1,k) + T(n-1,k-1) + binomial(n-1,k-1) for 2 <= k <= n with (n,k) <> (2,2).
T(n,k) = 2*T(n-1,k) - T(n-2,k) - T(n-2,k-2) + 2*T(n-1,k-1) - 2*T(n-2,k-1) for 3 <= k <= n with (n,k) <> (3,3).
T(n,1) = n - 1 for n >= 2.
T(n,2) = A002522(n-1) for n >= 2.
T(n,3) = A164845(n-3) for n >= 3.
T(n,4) = A332697(n-3) for n >= 4.
T(n,n) = n for n >= 1.
T(n,n-1) = A028387(n-2) for n >= 2. (End)
EXAMPLE
Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins:
1;
1, 2;
2, 5, 3;
3, 10, 11, 4;
4, 17, 27, 19, 5;
5, 26, 54, 56, 29, 6;
6, 37, 95, 130, 100, 41, 7;
...
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Feb 15 2007
STATUS
approved