OFFSET
0,5
COMMENTS
Depending on some fixed integer m there is a family of number triangles T(m; n,k) for 0 <= k <= n with entries: T(m; n,0) = 1 for n >= 0 and T(m; n,k) = binomial(n-1,k-1)*(m*k*(n-k) + n)/k for 0 < k <= n.
Further properties: T(m; n,n) = 1 for n >= 0; T(m; n,k) = T(m; n,n-k) for 0 <= k <= n; T(m; 2*n,n) = A000108(n)*A086270(m,n+1) for n >= 0 and m > 0.
T(m; n,k) = T(m; n-1,k) + T(m; n-1,k-1) + m*binomial(n-2,k-1) for 0 < k < n.
G.f. of column k: (1 + m*k*x) * x^k / (1 - x)^(k+1).
G.f.: A(x, t) = (1 - (1+x)*t + m*x*t^2) / (1 - (1+x)*t)^2.
T(m; n,k) = [x^k] (1 + (m*n - m + 2)*x + x^2) * (1 + x)^(n-2) for n > 0.
FORMULA
T(n, n) = 1; T(n, k) = T(n, n-k).
T(2*n, n) = (n+1)^2 * A000108(n).
T(n, k) = T(n-1, k) + T(n-1, k-1) + 2 * binomial(n-2,k-1) for 0 < k < n.
G.f. of column k: (1 + 2*k*x) * x^k / (1 - x)^(k+1).
G.f.: A(x,t) = (1 - (1 + x) * t + 2 * x * t^2) / (1 - (1 + x) * t)^2.
T(n,k) = [x^k] (1 + 2 * n * x + x^2) * (1 + x)^(n-2) for n > 0.
EXAMPLE
Triangle T(n, k) for 0 <= k <= n starts:
n\k : 0 1 2 3 4 5 6 7 8 9
=================================================
0 : 1
1 : 1 1
2 : 1 4 1
3 : 1 7 7 1
4 : 1 10 18 10 1
5 : 1 13 34 34 13 1
6 : 1 16 55 80 55 16 1
7 : 1 19 81 155 155 81 19 1
8 : 1 22 112 266 350 266 112 22 1
9 : 1 25 148 420 686 686 420 148 25 1
etc.
MATHEMATICA
Flatten[Table[Join[{1}, Table[Binomial[n-1, k-1](2*k*(n-k) + n)/k, {k, n}]], {n, 0, 10}]] (* Stefano Spezia, Jan 06 2022 *)
CROSSREFS
KEYWORD
AUTHOR
Werner Schulte, Jan 02 2022
STATUS
approved