A124376 Number triangle with column k generated by x^k*(1+2*k*x+C(k,2)*x^2)/(1-x)^(k+1).

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 19, 10, 1, 1, 13, 37, 37, 13, 1, 1, 16, 61, 92, 61, 16, 1, 1, 19, 91, 185, 185, 91, 19, 1, 1, 22, 127, 326, 440, 326, 127, 22, 1, 1, 25, 169, 525, 896, 896, 525, 169, 25, 1, 1, 28, 217, 792, 1638, 2072, 1638, 792, 217, 28, 1

Offset: 0

Author

Paul Barry, Oct 28 2006

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins
  1,
  1,  1,
  1,  4,  1,
  1,  7,  7,   1,
  1, 10, 19,  10,   1,
  1, 13, 37,  37,  13,  1,
  1, 16, 61,  92,  61, 16,  1,
  1, 19, 91, 185, 185, 91, 19, 1
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Crossrefs

Columns include A016777, A003215, A096000.

Cf. A158920.

Programs

Mathematica

A124376[n_, k_] := Sum[Binomial[k, k-j]*Binomial[n-j, k]*Binomial[2, j], {j, 0, n}];
Table[A124376[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Feb 21 2025 *)

PARI

C(i,j) =binomial(i,j);
T(n,k) = if (k<=n, sum(j=0, n, C(k,k-j)*C(n-j,k)*C(2,j)));
row(n) = vector(n+1, k, T(n,k-1));
for (n=0, 10, print(row(n))) \\ Michel Marcus, Feb 19 2025

Formula

T(n,k) = Sum_{j=0..n} C(k,k-j)*C(n-j,k)*C(2,j)*[k<=n].

T(n,k) = T(n,n-k).

Extensions

More terms from Michel Marcus, Feb 19 2025