A110180 Triangle of generalized central trinomial coefficients.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 5, 1, 1, 1, 19, 13, 7, 1, 1, 1, 51, 49, 19, 9, 1, 1, 1, 141, 161, 91, 25, 11, 1, 1, 1, 393, 581, 331, 145, 31, 13, 1, 1, 1, 1107, 2045, 1441, 561, 211, 37, 15, 1, 1, 1, 3139, 7393, 5797, 2841, 851, 289, 43, 17, 1, 1

Offset: 0

Author

Paul Barry, Jul 14 2005

Keywords

• easy
• nonn
• tabl

Comments

Rows sums are A110181. Diagonal sums are A110182. Columns include central trinomial coefficients A002426, A084601, A084603, A084605, A098264. T(n,k) = central coefficient (1 + x + kx^2)^n.

Examples

			Rows begin
  1;
  1,  1;
  1,  1,  1;
  1,  3,  1,  1;
  1,  7,  5,  1,  1;
  1, 19, 13,  7,  1,  1;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Programs

Mathematica

T[n_, 0] := 1; T[n_, k_] := Sum[Binomial[n - k, j]*Binomial[n - k - j, j]*k^j, {j, 0, Floor[(n - k)/2]}]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Mar 05 2017 *)

Formula

Number triangle T(n, k) = Sum_{j=0..floor((n-k)/2)} C(n-k, j)*C(n-k-j, j)*k^j.