A350512 Triangle read by rows with T(n,0) = 1 for n >= 0 and T(n,k) = binomial(n-1,k-1)*(2*k*(n-k) + n)/k for 0 < k <= n.

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 18, 10, 1, 1, 13, 34, 34, 13, 1, 1, 16, 55, 80, 55, 16, 1, 1, 19, 81, 155, 155, 81, 19, 1, 1, 22, 112, 266, 350, 266, 112, 22, 1, 1, 25, 148, 420, 686, 686, 420, 148, 25, 1, 1, 28, 189, 624, 1218, 1512, 1218, 624, 189, 28, 1

Offset: 0

Author

Werner Schulte, Jan 02 2022

Keywords

• nonn
• easy
• tabl

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.

Special cases: m=0 (A007318), m=1 (A103450), and m=2 (this triangle).

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.

Examples

			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.
		

Crossrefs

Row sums are A057711(n+1).

Cf. A000108, A007318, A086270, A103450.

Programs

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 *)

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.